Sesion 13 - Laboratorio de Analisis de Factores
Juan Carlos Martinez-Ovando
Departamento Academico de Actuaria
Primavera 2019
1 Datos
Consideremos de nuevo los datos de tipos de cambio que hemos venido empleando en el curso.
data <- read.csv("est46114_s06_data.csv")
data <- as.data.frame(data)
data <- as.matrix(data)T <- nrow(data); p <- ncol(data)-2datats <- ts(data[,c(3,4,p+2)],start=c(1970, 1), end=c(2010, 12), frequency=12)
plot(datats)
2 Modelo de factores
El modelo de factores considera que tiempo-a-tiempo (i.e. para cada \(t\) mes), el bloque de los \(80\) tipos de cambio, o de los \(3\) tipos de cambio (caso reducido para ‘Canada’,‘Mexico’,‘Zambia’), denotados por \(\boldsymbol{X}_t\), pueden modelarse como \[
\boldsymbol{X}_t \sim \text{N}_p(\boldsymbol{x}|\boldsymbol{\theta},\boldsymbol{\Omega}),
\] donde
\(\boldsymbol{\theta}\) es la media general en \(t\), y
\(\boldsymbol{\Omega}\) es la matriz de varianzas y covarianzas de los \(p\) tipos de cambio.
2.1 Supuestos
Asi, el modelo implicitamente supone:
Homogeneidad, i.e. $ _p(|,)$ en su ‘forma estructural’ y ‘parametros’ sea el mismo para todo \(t\).
Invarainza al orden, i.e. \[ \mathbb{P}(\boldsymbol{x}_1,\ldots,\boldsymbol{x}_t)=\mathbb{P}(\boldsymbol{x}_{\sigma(1)},\ldots,\boldsymbol{x}_{\sigma(t)}). \] Induciendo que el ‘modelo lea’ las observaciones independientemente o condicionalmente independiente en \(t\).
2.2 Consideraciones
Hemos visto que los supuestos (1) y (2) no son sustentables en este conjunto de datos.
2.2.1 Independencia
Si los datos tuviesen proveyeran informacion acerca de la independencia en \(t\), tendriamos que ‘muestralmente’, las correlaciones de los datos en \(t\) respecto a tiempos previos (rezagos), seria ‘nula’.
La funcion de autocorrelacion nos brinda una herramienta para inspeccionar este supuesto.
acf(datats[,"Canada"])
acf(datats[,"Mexico"])
acf(datats[,"Zambia"])
2.2.2 Tendencia
Si el modelo homogeneo en tendencia (medias, o \(\boldsymbol{\theta}_t\)s) fuese admisible para el conjunto de datos, tendriamos que para todo orden de observaciones el comportamiento de las medias por ‘subsegmentos’ de datos seria muy semejante. En la grafica vemos que en este caso, esto no se cumple.
datats <- ts(data[,3:(p+2)],start=c(1970, 1), end=c(2010, 12), frequency=12)
Y <- T/12
years <- seq(1970,2010,1)
meansts <- NA * datats[c(1:41),]
t <- 1
for(t in 1:Y){
meansts[t, ] <- colMeans(data[which(data[,"Year"]==years[t]),3:(p+2)])
}plot(meansts[,"Canada"])
plot(meansts[,"Mexico"])
plot(meansts[,"Zambia"])
rep.row<-function(x,n){
matrix(rep(x,each=n),nrow=n)
}
rep.col<-function(x,n){
matrix(rep(x,each=n), ncol=n, byrow=TRUE)
}Corrigiendo por medias temporales, i.e. relajando el supesto que \[ \mathbb{E}(\boldsymbol{X}_t)=\boldsymbol{\theta}, \] para todo \(t\), por el siguiente, \[ \mathbb{E}(\boldsymbol{X}_t)=\boldsymbol{\theta}_a, \] donde \(a\) corresponde al anio calendario correspondiente a cada \(t\), observamos un efecto de correccion en tendencia.
datatsc <- datats
t <- 1
for(t in 1:Y){
datatsc[which(data[,"Year"]==years[t]),] <- datats[which(data[,"Year"]==years[t]),] - rep.row(meansts[t, ],12)
}plot(datatsc[,c("Canada","Mexico","Zambia")])
2.2.3 Independencia (revisada)
El efecto de correccion en tendencia implica necesariamente un efecto en terminos de la estructura de dependencia intrinseca (autocorrelacion).
acf(datatsc[,"Canada"])
acf(datatsc[,"Mexico"])
acf(datatsc[,"Zambia"])
2.3 Invarianza al orden
if(!require("gtools")){install.packages("gtools")}## Loading required package: gtoolslibrary("gtools")
t.index <- c(1:T)
set.seed(1)
t.sample1 <- sample(t.index)
t.sample2 <- sample(t.index)
t.sample3 <- sample(t.index)2.3.0.1 Datos originales
plot(ts(datatsc[t.index,c("Canada","Mexico","Zambia")],frequency=12))
#### Primera permutacion
plot(ts(datatsc[t.sample1,c("Canada","Mexico","Zambia")],frequency=12))
#### Segunda permutacion
plot(ts(datatsc[t.sample2,c("Canada","Mexico","Zambia")],frequency=12))
2.3.0.2 Tercera permutacion
plot(ts(datatsc[t.sample3,c("Canada","Mexico","Zambia")],frequency=12))
2.4 Implementacion
if(!require('MCMCpack')){install.packages("MCMCpack")}## Loading required package: MCMCpack## Loading required package: coda## Loading required package: MASS## ##
## ## Markov Chain Monte Carlo Package (MCMCpack)## ## Copyright (C) 2003-2019 Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park## ##
## ## Support provided by the U.S. National Science Foundation## ## (Grants SES-0350646 and SES-0350613)
## ####
## Attaching package: 'MCMCpack'## The following objects are masked from 'package:gtools':
##
## ddirichlet, rdirichletlibrary("MCMCpack")
datatsc.3 <- as.data.frame(datatsc[t.index,c("Canada","Mexico","Zambia")])
fa.posterior <- MCMCfactanal(~Canada+Mexico+Zambia,
factors=1,
lambda.constraints=list(Mexico1=c(1,0),
Zambia1=c(1,0)),
verbose=0, store.scores=FALSE,
a0=1, b0=0.15,
data=datatsc.3,
burnin=50, mcmc=150, thin=1)
fa.posterior## Markov Chain Monte Carlo (MCMC) output:
## Start = 51
## End = 200
## Thinning interval = 1
## LambdaCanada_1 LambdaMexico_1 LambdaZambia_1 PsiCanada PsiMexico
## [1,] -1.313440e-01 -0.8038129 -0.2816096 1.0061454 0.4116594
## [2,] -1.570016e-01 -0.7579327 -0.3039861 0.9361589 0.3959228
## [3,] -1.614056e-01 -0.7879895 -0.2256421 0.8798418 0.3875877
## [4,] -1.680800e-01 -0.8266690 -0.1423309 0.8771918 0.3497977
## [5,] -1.674268e-01 -0.8235891 -0.1972571 0.9725896 0.3506025
## [6,] -7.691657e-02 -0.7753728 -0.2506273 0.9330723 0.2983826
## [7,] -7.955401e-02 -0.8268506 -0.3130412 0.8937084 0.3833808
## [8,] 1.069530e-02 -0.7540654 -0.3570257 1.0461590 0.3153575
## [9,] -2.546568e-02 -0.7507844 -0.3059460 1.0892251 0.4065127
## [10,] 1.595217e-03 -0.7802823 -0.2139933 1.0323348 0.4558982
## [11,] -1.476444e-01 -0.7833344 -0.2675254 0.9937036 0.3797556
## [12,] -1.806595e-01 -0.7563406 -0.2508719 0.9270239 0.3797318
## [13,] -1.002755e-01 -0.7602890 -0.2597754 0.9555030 0.4363572
## [14,] -5.932690e-02 -0.7225603 -0.2714403 0.9803167 0.5128466
## [15,] -1.517905e-01 -0.7110268 -0.3177422 0.9602993 0.4525728
## [16,] -1.135719e-01 -0.6442257 -0.3620564 0.9689639 0.5089521
## [17,] -1.688329e-01 -0.6683868 -0.3208790 1.0387238 0.5797079
## [18,] -2.325099e-01 -0.6685861 -0.2922009 1.0936516 0.6332812
## [19,] -2.388811e-01 -0.6050263 -0.2730728 0.9622687 0.5713836
## [20,] -2.368702e-01 -0.6232182 -0.2488101 0.9746449 0.6138756
## [21,] -1.724804e-01 -0.5906095 -0.1687467 0.8820698 0.5279362
## [22,] -1.059608e-01 -0.6103843 -0.2437824 1.0526378 0.5774108
## [23,] -1.081365e-01 -0.6252663 -0.2151890 1.0429392 0.6938378
## [24,] -2.885507e-01 -0.5991988 -0.2432485 0.9378205 0.6576875
## [25,] -3.032972e-01 -0.5908190 -0.2970895 1.0386433 0.6281160
## [26,] -2.357407e-01 -0.6287761 -0.3321617 0.9129389 0.6336322
## [27,] -1.749892e-01 -0.6327618 -0.3414023 0.8865521 0.5184952
## [28,] -1.306157e-01 -0.5769780 -0.3203061 0.9416544 0.6357294
## [29,] -1.006285e-01 -0.4986698 -0.3574179 0.9608514 0.6208821
## [30,] -2.191599e-01 -0.5786390 -0.3617068 0.9594067 0.7527005
## [31,] -2.697413e-01 -0.5614428 -0.3557484 0.9750482 0.7097464
## [32,] -1.994997e-01 -0.5647001 -0.3617366 0.8070104 0.7523981
## [33,] -1.866581e-01 -0.6063550 -0.3755794 0.9279500 0.8036021
## [34,] -1.901035e-01 -0.5919677 -0.2686846 0.8962285 0.7149832
## [35,] -1.002256e-01 -0.6099510 -0.3414800 0.9886101 0.6434619
## [36,] -2.192024e-01 -0.6341711 -0.2659336 1.0018529 0.6627565
## [37,] -1.969789e-01 -0.5223273 -0.3100717 0.8675846 0.6441029
## [38,] -2.074480e-01 -0.5225189 -0.4567816 0.9437811 0.7864313
## [39,] -1.005176e-01 -0.5804788 -0.3778731 1.0843592 0.7024209
## [40,] -1.349020e-01 -0.6031803 -0.3822221 1.0054097 0.6922420
## [41,] -9.354188e-02 -0.6073132 -0.2473797 1.0390408 0.6091122
## [42,] -3.712989e-02 -0.6624150 -0.2607910 0.9365328 0.6212531
## [43,] -1.428434e-01 -0.6142774 -0.2821468 0.9547870 0.5463931
## [44,] -8.660184e-02 -0.6400922 -0.2701131 1.0106911 0.5063704
## [45,] -5.362241e-02 -0.6871810 -0.2958875 0.9190562 0.5746872
## [46,] -6.375656e-02 -0.7577042 -0.3497381 0.8958022 0.5283322
## [47,] -1.381138e-01 -0.7947216 -0.3055979 0.9915012 0.5127269
## [48,] -1.125345e-01 -0.7608732 -0.2967274 1.0020842 0.5473486
## [49,] -7.343879e-02 -0.6733398 -0.3029775 1.0499278 0.4710796
## [50,] -1.737512e-01 -0.7025193 -0.3215639 0.9840202 0.5755468
## [51,] -1.559548e-01 -0.6834554 -0.3278621 0.9401004 0.6172367
## [52,] -1.899308e-01 -0.6561680 -0.2766643 0.8863264 0.5176300
## [53,] -1.609041e-01 -0.6287876 -0.3489765 0.9936052 0.6477754
## [54,] -1.658593e-01 -0.6293869 -0.3423813 0.9302969 0.5825986
## [55,] -3.077530e-01 -0.6333443 -0.2567675 0.9490136 0.6015178
## [56,] -2.511484e-01 -0.7089937 -0.3093567 0.9602045 0.5831954
## [57,] -1.112871e-01 -0.6566115 -0.2281089 0.8235235 0.5349589
## [58,] -1.792509e-01 -0.6984216 -0.2313344 1.1014390 0.5172283
## [59,] -3.945610e-02 -0.7244162 -0.2582507 0.9326448 0.5051566
## [60,] -5.749231e-06 -0.7622402 -0.2763557 1.0622995 0.5665040
## [61,] -4.099813e-02 -0.8475316 -0.3182463 0.9974873 0.5262669
## [62,] -1.103989e-01 -0.7801294 -0.3307596 1.0088856 0.4252963
## [63,] -1.260015e-01 -0.7434451 -0.2884836 0.8865973 0.3748638
## [64,] -7.381889e-02 -0.7695053 -0.2492356 0.9519695 0.4074981
## [65,] -1.081957e-01 -0.7135481 -0.2976787 0.8576031 0.3767433
## [66,] -1.017797e-01 -0.7634943 -0.3130865 1.1135997 0.3936456
## [67,] -1.021198e-01 -0.7766113 -0.2325455 0.9270748 0.3795737
## [68,] -8.478626e-02 -0.8112321 -0.2204310 0.8834349 0.4068485
## [69,] -1.404440e-01 -0.8109807 -0.2271326 0.9272189 0.3739250
## [70,] -1.936729e-01 -0.7515774 -0.2143876 0.9148333 0.4229171
## [71,] -1.394633e-01 -0.7484795 -0.2287667 1.0004243 0.3903694
## [72,] -1.286486e-01 -0.7308400 -0.2279740 0.9935352 0.3219727
## [73,] -9.846330e-02 -0.8012564 -0.2354420 1.0648581 0.3581573
## [74,] -1.193247e-01 -0.8148217 -0.2446225 0.9996250 0.3333158
## [75,] -1.304124e-01 -0.8367927 -0.2134898 0.8913346 0.3896981
## [76,] -1.345060e-01 -0.8240153 -0.3176188 1.0182160 0.3270810
## [77,] -7.990427e-02 -0.8041979 -0.2318064 1.0237494 0.3333583
## [78,] -3.214348e-02 -0.7473483 -0.1818638 0.9971271 0.3416685
## [79,] -3.892430e-02 -0.7474312 -0.2412871 1.0150209 0.3552439
## [80,] -6.789114e-02 -0.7950808 -0.2188696 1.0157041 0.3737743
## [81,] -1.614006e-01 -0.7999718 -0.3323355 1.0065656 0.3851088
## [82,] -1.824901e-01 -0.8249488 -0.2505241 0.9943472 0.3582281
## [83,] -2.363117e-01 -0.7787434 -0.2864547 0.8602734 0.3320010
## [84,] -2.052754e-01 -0.7834870 -0.2810964 0.9245595 0.3101502
## [85,] -2.805652e-01 -0.7982204 -0.1859628 0.9487280 0.3037625
## [86,] -1.939421e-01 -0.8271568 -0.1790840 1.0709482 0.3301610
## [87,] -1.505755e-01 -0.8371194 -0.1631539 1.0643666 0.3202306
## [88,] -9.323318e-02 -0.8565474 -0.2222155 0.9827947 0.2780590
## [89,] -6.633465e-02 -0.7974952 -0.2741091 0.9688497 0.2907187
## [90,] -1.248921e-01 -0.8612397 -0.2579029 0.9606209 0.2966333
## [91,] -8.603955e-02 -0.8629897 -0.2734082 1.0815317 0.2954396
## [92,] -1.895134e-01 -0.8982412 -0.2731863 1.0317103 0.2357103
## [93,] -1.444651e-01 -0.8766824 -0.2521674 1.1019412 0.2258737
## [94,] -1.804944e-01 -0.8357706 -0.1931679 0.9955691 0.2391250
## [95,] -1.300700e-01 -0.8841026 -0.2028720 0.8875567 0.2610720
## [96,] -4.155302e-02 -0.8638341 -0.2932436 0.9069720 0.2214591
## [97,] -1.066283e-01 -0.8934176 -0.2637394 1.0435918 0.2505003
## [98,] -2.108477e-02 -0.8396800 -0.2967805 0.9771144 0.2549567
## [99,] -1.189242e-01 -0.8269176 -0.2348267 0.9379920 0.2538293
## [100,] -1.531149e-01 -0.8288699 -0.1736272 0.9897010 0.2729557
## [101,] -7.343503e-02 -0.8586443 -0.2671735 1.0584125 0.2488753
## [102,] -1.262218e-01 -0.8917072 -0.2734164 0.9193474 0.2667662
## [103,] -2.064438e-01 -0.9166786 -0.2580935 0.9495710 0.2312843
## [104,] -1.302428e-01 -0.8724548 -0.2095246 1.0573440 0.2623110
## [105,] -1.774327e-01 -0.8887345 -0.2148600 0.9689191 0.2439879
## [106,] -2.604726e-01 -0.9080614 -0.2259497 1.0713583 0.2203756
## [107,] -1.574909e-01 -0.8934616 -0.2749616 0.9571631 0.2366237
## [108,] -1.945140e-01 -0.9024526 -0.2453184 0.9332897 0.2784568
## [109,] -1.966169e-01 -0.9440960 -0.2863667 0.9654002 0.3109694
## [110,] -4.704324e-02 -0.9318366 -0.2786000 0.9367688 0.3156816
## [111,] -1.510234e-01 -0.8640621 -0.2818296 1.0298093 0.2734327
## [112,] -1.154050e-01 -0.8722377 -0.1393866 1.0449739 0.2562684
## [113,] -1.126535e-01 -0.9027718 -0.2592732 0.9233598 0.2533854
## [114,] -1.189043e-01 -0.8558491 -0.2594629 1.0682758 0.2248855
## [115,] -1.289682e-01 -0.8030648 -0.2982030 0.9807193 0.2543455
## [116,] -7.538957e-02 -0.8541609 -0.2541948 1.1411476 0.2497804
## [117,] -1.041427e-02 -0.8687691 -0.2238398 0.9765312 0.2787748
## [118,] -2.440391e-02 -0.8787642 -0.1779331 1.0956801 0.2317387
## [119,] -1.243644e-01 -0.8421912 -0.2167075 0.9727927 0.2603127
## [120,] -5.717118e-02 -0.8603274 -0.2114056 0.9788596 0.2498312
## [121,] -4.642273e-02 -0.8931270 -0.2393605 0.9179936 0.2508926
## [122,] -6.934878e-02 -0.8494479 -0.2084370 0.9394036 0.2717849
## [123,] -5.017460e-02 -0.8332300 -0.1863026 1.0140950 0.2674003
## [124,] -2.034862e-01 -0.8238387 -0.1837118 0.9704464 0.2672748
## [125,] -1.696534e-01 -0.8888636 -0.2731107 1.1119611 0.3077279
## [126,] -2.159597e-01 -0.8826986 -0.3340160 0.8457513 0.3167757
## [127,] -1.573342e-01 -0.8559097 -0.3328415 0.9603885 0.3021574
## [128,] -1.069889e-01 -0.8331950 -0.3130323 1.0457178 0.2934011
## [129,] -1.516062e-01 -0.8678489 -0.3002747 0.9778591 0.2754519
## [130,] -1.066895e-01 -0.9016105 -0.2985903 0.9170580 0.2756804
## [131,] -1.533520e-01 -0.9096727 -0.2326407 1.1001621 0.2728300
## [132,] -1.115418e-01 -0.9191171 -0.2250652 0.9614081 0.2295588
## [133,] -1.420211e-01 -0.8818978 -0.3099841 0.9816122 0.2254461
## [134,] -1.594246e-01 -0.8516391 -0.2522270 0.9899600 0.2271001
## [135,] -8.371003e-02 -0.8769231 -0.2455377 0.9558094 0.2105616
## [136,] -8.115337e-02 -0.8952963 -0.2760660 0.9957728 0.2121907
## [137,] -2.610045e-01 -0.9152883 -0.3146934 0.9267679 0.2515472
## [138,] -1.224313e-01 -0.8735643 -0.2451977 0.9661833 0.2663440
## [139,] -1.948237e-01 -0.9253350 -0.2978792 0.9868680 0.2542990
## [140,] -1.645429e-01 -0.9240053 -0.2841008 0.9279769 0.2860417
## [141,] -1.165529e-01 -0.8783499 -0.1670780 1.1467915 0.3062565
## [142,] -2.785001e-01 -0.8313462 -0.1818295 1.0068730 0.2975372
## [143,] -1.868471e-01 -0.8444385 -0.3091613 0.9361075 0.2972175
## [144,] -3.257599e-02 -0.8621752 -0.3017831 0.9548435 0.2741142
## [145,] -1.025659e-01 -0.8801877 -0.2387719 0.9168576 0.2615302
## [146,] -6.654101e-02 -0.8737189 -0.2421176 0.9381385 0.2720359
## [147,] -1.790850e-01 -0.8413226 -0.2090160 1.1239113 0.2752455
## [148,] -1.522042e-01 -0.8109911 -0.2447516 1.0094764 0.3120314
## [149,] -9.590794e-02 -0.8449021 -0.1921708 0.9690455 0.2825912
## [150,] -1.329869e-01 -0.8125215 -0.1301149 0.9733932 0.2621129
## PsiZambia
## [1,] 0.9954873
## [2,] 1.0070362
## [3,] 1.0516263
## [4,] 0.9532203
## [5,] 0.9044722
## [6,] 0.9326477
## [7,] 0.8615549
## [8,] 0.9336869
## [9,] 0.9683080
## [10,] 1.0592374
## [11,] 1.0108144
## [12,] 1.0153966
## [13,] 0.9147160
## [14,] 0.9379959
## [15,] 0.9355091
## [16,] 0.8692317
## [17,] 0.8701564
## [18,] 0.8482479
## [19,] 0.9563067
## [20,] 0.9721528
## [21,] 0.8854596
## [22,] 0.8447454
## [23,] 0.9042113
## [24,] 0.8544734
## [25,] 1.0202484
## [26,] 0.9032024
## [27,] 0.8456380
## [28,] 0.8577884
## [29,] 0.8353255
## [30,] 0.8546962
## [31,] 0.8581595
## [32,] 0.9264553
## [33,] 0.9564432
## [34,] 0.8896792
## [35,] 0.8893527
## [36,] 0.8444730
## [37,] 0.8553654
## [38,] 0.8178582
## [39,] 0.8688949
## [40,] 0.8919509
## [41,] 0.8775137
## [42,] 1.0250008
## [43,] 0.8426380
## [44,] 0.9157439
## [45,] 0.8800744
## [46,] 0.8549704
## [47,] 0.8669941
## [48,] 0.9916016
## [49,] 0.8879934
## [50,] 0.8706642
## [51,] 0.8728135
## [52,] 0.8586143
## [53,] 0.9154176
## [54,] 0.8964398
## [55,] 0.9295307
## [56,] 0.8601873
## [57,] 0.7869318
## [58,] 0.9410080
## [59,] 0.8597775
## [60,] 0.9493921
## [61,] 0.9446040
## [62,] 0.9469455
## [63,] 0.9537974
## [64,] 0.8690808
## [65,] 0.8848051
## [66,] 0.9601448
## [67,] 0.8527361
## [68,] 1.0435621
## [69,] 0.9884363
## [70,] 0.9397550
## [71,] 0.8993054
## [72,] 1.0605649
## [73,] 0.8918531
## [74,] 0.8743727
## [75,] 1.0309277
## [76,] 0.9828107
## [77,] 0.9252888
## [78,] 0.9770418
## [79,] 0.9028271
## [80,] 0.9302682
## [81,] 0.9615108
## [82,] 0.9258265
## [83,] 0.9225817
## [84,] 0.8744992
## [85,] 1.0142766
## [86,] 1.0306481
## [87,] 1.0891932
## [88,] 0.9930969
## [89,] 1.1160303
## [90,] 1.0070212
## [91,] 0.9201742
## [92,] 0.8833454
## [93,] 0.9885911
## [94,] 1.0343708
## [95,] 0.8939333
## [96,] 1.0102840
## [97,] 0.8639778
## [98,] 0.8783241
## [99,] 0.9714797
## [100,] 0.8965222
## [101,] 0.8833697
## [102,] 0.9301872
## [103,] 0.9453331
## [104,] 0.8468230
## [105,] 0.9617060
## [106,] 0.9218789
## [107,] 0.9446461
## [108,] 1.0103551
## [109,] 0.9514989
## [110,] 0.9294979
## [111,] 0.9604210
## [112,] 0.9589084
## [113,] 0.9151665
## [114,] 0.9713864
## [115,] 0.9745754
## [116,] 0.9635867
## [117,] 1.1077937
## [118,] 0.9674457
## [119,] 1.0150687
## [120,] 0.9676572
## [121,] 0.8978634
## [122,] 0.8401712
## [123,] 0.8874293
## [124,] 1.0031673
## [125,] 0.9510003
## [126,] 0.8455523
## [127,] 0.9084703
## [128,] 0.9190738
## [129,] 0.9280782
## [130,] 0.9333786
## [131,] 1.0343486
## [132,] 0.9162654
## [133,] 1.0132667
## [134,] 0.9907024
## [135,] 1.0044713
## [136,] 1.0375815
## [137,] 0.9628848
## [138,] 0.9025556
## [139,] 1.0078061
## [140,] 0.8685727
## [141,] 1.0256659
## [142,] 0.9123188
## [143,] 0.9378514
## [144,] 0.8856493
## [145,] 0.9027066
## [146,] 0.9548190
## [147,] 1.0734965
## [148,] 0.9383125
## [149,] 1.0101698
## [150,] 1.0216437
## attr(,"constraints")
## attr(,"constraints")$Mexico1
## [1] 1 0
##
## attr(,"constraints")$Zambia1
## [1] 1 0
##
## attr(,"n.manifest")
## [1] 3
## attr(,"n.factors")
## [1] 1summary(fa.posterior)##
## Iterations = 51:200
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 150
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## LambdaCanada_1 -0.1351 0.06616 0.005402 0.009056
## LambdaMexico_1 -0.7755 0.10815 0.008830 0.065804
## LambdaZambia_1 -0.2665 0.05541 0.004524 0.008186
## PsiCanada 0.9781 0.06602 0.005390 0.005390
## PsiMexico 0.3995 0.15549 0.012695 0.108898
## PsiZambia 0.9351 0.06511 0.005316 0.008709
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## LambdaCanada_1 -0.2791 -0.1787 -0.1305 -0.09331 -0.01815
## LambdaMexico_1 -0.9205 -0.8619 -0.8034 -0.69945 -0.56380
## LambdaZambia_1 -0.3658 -0.3037 -0.2666 -0.22827 -0.16600
## PsiCanada 0.8595 0.9340 0.9731 1.01479 1.11241
## PsiMexico 0.2239 0.2722 0.3457 0.52432 0.72527
## PsiZambia 0.8420 0.8837 0.9299 0.97643 1.064122.4.0.1 Comentarios sobre la implementacion
Estamos considerando que los niveles medios \((\boldsymbol{\theta}_a)_{a\geq 1970}\) son “fijos”/“conocidos”para el modelo, aunque estos los hayamos calculado con los datos.
Los coeficientes (
LambdaCanada_1,LambdaMexico_1,LambdaZambia_1) corresponden al vector de cargas (unico, en este caso) contenido en la matriz \(\Lambda\).Los coeficientes (
PisCanada,PsiMexico,PsiZambia) corresponden a las varianzas contenidas en la diagonal de la matriz \(\Sigma\).
2.5 Diagrama de dependencia
Diagrama de dependencia del modelo de factores estatico
3 Especificacion dinamica en factores
En este caso, es de interes recuperar informacion respecto a la tendencia observada en los datos \((\boldsymbol{X}_t)_{t\geq 1}\) en terminos de los factores latentes, \((\boldsymbol{f}_t)_{t\geq 1}\) y de un componente en niveles \((\boldsymbol{\theta}_t)_{t\geq 1}\).
Asi, para los datos \((p\times 1)\)-dimensionales, el modelo es de la forma \[\begin{eqnarray} \boldsymbol{X}_t|\boldsymbol{\theta}_t,\boldsymbol{f}_t & \sim & N_p(\boldsymbol{x}|\boldsymbol{\theta}_t+\Lambda \boldsymbol{f}_t,\Sigma)\nonumber \\ \end{eqnarray}\] para \(t\geq 1\), donde \(\boldsymbol{\theta}_t\) es un vector \((p\times 1)\)-dimensional, \(\boldsymbol{f}_t\) es un vector \((k\times 1)\)-dimensional (con \(k<<p\)), tales que \[\begin{eqnarray} \boldsymbol{\theta}_t|\boldsymbol{\theta}_{t-1} & \sim & N_p(\boldsymbol{\theta}|\boldsymbol{f}_{t-1},\boldsymbol{W}) \nonumber, \\ \boldsymbol{f}_t|\boldsymbol{f}_{t-1} & \sim & N_k(\boldsymbol{f}|\Lambda \boldsymbol{f}{t-1},\boldsymbol{G}) \nonumber, \end{eqnarray}\] donde
\(\Lambda\) es la matriz \((p\times k)\)-dimensional de cargas de factores,
\(\boldsymbol{W}\) es una matriz de varianzas y covarianzas (tipicamente conocida o preespecificada) de dimension \((p\times p)\) y
\(\boldsymbol{G}\) es una matriz diagonal de varianzas de dimension \((k\times k)\) (tambien, usualmente, fija o predefinida).
3.1 Parametros
Los parametros del modelo quedan entonces especificados como \[ \left(\Lambda,\boldsymbol{\Sigma}\right), \] siendo las variables latentes \[ \left((\boldsymbol{f}_{t})_{t\geq 1}\right). \] Las condiciones de identificabilidad del modelo son iguales a las tradicionales, respecto a \(\Lambda\).
3.2 Observaciones
El parametro \(\boldsymbol{\theta}_t\) brinda una lectura acerca de la tendencia de los datos, en \(t\), de manera separada para las \(p\) dimensiones.
El parametro \(\boldsymbol{f}_t|\boldsymbol{x}_t\) brinda informacion acerca del componente coyuntural de correlacion subyacente a \(\boldsymbol{x}_t\) en \(t\).
El componente de tendencia coyuntural agregado es, en este caso, \[\begin{eqnarray} \mathbb{E}(\boldsymbol{x}|\boldsymbol{f}_t) & = & \boldsymbol{\theta}_t + \Lambda \boldsymbol{f}_t, \nonumber \end{eqnarray}\] mientras que el compoente de tendencia coyuntural marginalizado es, \[\begin{eqnarray} \mathbb{E}(\boldsymbol{x}|\boldsymbol{f}_{t-1}) & = & \int \boldsymbol{x}_t N_p(\boldsymbol{x}_t|\boldsymbol{\theta}_t+\Lambda\boldsymbol{t}_t)N_k(\boldsymbol{f}_t|\boldsymbol{f}_{t-1},\boldsymbol{G})\text{d}\boldsymbol{f}_t \nonumber \\ & = & \boldsymbol{\theta}_t + \Lambda \boldsymbol{f}_{t-1}. \nonumber \end{eqnarray}\]
Esta ultima es una media que hereda un componente del periodo previo. A diferencia del modelo estatico original, el efecto de \(\boldsymbol{f}_t\) no se desvanece necesariamente.
3.3 Asociaciones
El modelo anterior puede verse como un caso particular del modelo state-space o dynamic linear model, donde el componente de estados es dual, i.e., \[ (\boldsymbol{\theta}_t,\boldsymbol{f}_t)_{t\geq 1}. \] Vean la referencia de West & Harrison en las lecturas complementarias.
Es de notar que el componente de reduccion de dimensionalidad incorporado induce a que la estimacion de parametros no pueda realizarse de manera analitica cerrada empleando el filtro de Kalman. En este caso, algorimos del tipo MCMC y relacionados son empleados.
3.4 Implementacion
if(!require("bayesdfa")){install.packages("bayesdfa")}## Loading required package: bayesdfa## Loading required package: Rcpplibrary("bayesdfa")
dfa.posterior <- fit_dfa(y = t(datatsc.3), iter = 150, chains = 1)## trying deprecated constructor; please alert package maintainer##
## SAMPLING FOR MODEL 'dfa' NOW (CHAIN 1).
## Chain 1:
## Chain 1: Gradient evaluation took 0.000451 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 4.51 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1:
## Chain 1:
## Chain 1: WARNING: There aren't enough warmup iterations to fit the
## Chain 1: three stages of adaptation as currently configured.
## Chain 1: Reducing each adaptation stage to 15%/75%/10% of
## Chain 1: the given number of warmup iterations:
## Chain 1: init_buffer = 11
## Chain 1: adapt_window = 57
## Chain 1: term_buffer = 7
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## Chain 1:
## Chain 1: Elapsed Time: 33.889 seconds (Warm-up)
## Chain 1: 23.484 seconds (Sampling)
## Chain 1: 57.373 seconds (Total)
## Chain 1:## Warning in sqrt(ess): NaNs produced## Inference for the input samples (1 chains: each with iter=75; warmup=37):
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5%
## x[1,1] 0.3 0.2 1.3 -1.9 -0.5 0.3 0.8 2.8
## x[1,2] 0.6 0.2 1.5 -2.1 -0.6 0.6 1.3 3.3
## x[1,3] 0.6 0.2 1.3 -1.5 -0.2 0.5 1.4 3.1
## x[1,4] 0.6 0.1 0.9 -1.1 0.0 0.6 1.1 2.2
## x[1,5] 0.4 0.1 1.0 -1.0 0.0 0.4 0.8 2.4
## x[1,6] -0.1 0.2 1.2 -2.5 -1.0 -0.1 0.7 2.0
## x[1,7] -0.4 0.0 1.6 -4.1 -0.9 -0.2 0.7 1.8
## x[1,8] -0.8 0.2 1.4 -3.6 -1.5 -0.6 0.1 1.3
## x[1,9] -1.0 0.2 1.4 -3.7 -1.9 -1.1 -0.2 1.3
## x[1,10] -0.7 0.1 1.2 -3.1 -1.3 -0.8 0.2 1.3
## x[1,11] -0.4 0.2 1.1 -2.2 -1.2 -0.5 0.4 1.7
## x[1,12] -0.2 0.2 1.1 -2.1 -1.0 -0.2 0.5 1.6
## x[1,13] 0.6 0.2 1.7 -1.8 -0.7 0.3 1.4 3.7
## x[1,14] 0.3 0.2 1.4 -1.5 -0.5 0.0 1.1 3.2
## x[1,15] 0.4 0.1 1.5 -2.2 -0.4 0.2 1.4 2.9
## x[1,16] 0.1 0.2 1.2 -2.1 -0.6 0.0 1.0 2.1
## x[1,17] 0.1 0.1 1.1 -1.5 -0.6 0.0 0.7 2.3
## x[1,18] 0.3 0.2 1.2 -2.4 -0.4 0.3 1.2 2.4
## x[1,19] -0.1 0.2 1.7 -3.8 -1.0 0.3 1.0 2.4
## x[1,20] -0.1 0.2 1.2 -2.2 -0.5 -0.1 0.5 2.4
## x[1,21] -0.1 0.2 1.2 -2.1 -0.9 -0.2 0.9 1.8
## x[1,22] -0.1 0.1 1.2 -1.8 -1.0 -0.3 0.5 2.6
## x[1,23] 0.2 0.2 1.8 -2.0 -1.0 -0.2 1.5 3.8
## x[1,24] -0.1 0.2 1.3 -2.2 -1.0 -0.1 0.7 2.2
## x[1,25] 0.5 0.2 1.3 -1.9 -0.3 0.4 1.0 3.2
## x[1,26] 0.6 0.1 1.2 -1.5 -0.1 0.6 1.1 2.8
## x[1,27] 0.3 0.1 1.0 -2.3 -0.3 0.3 1.0 2.1
## x[1,28] 0.2 0.1 1.0 -2.0 -0.1 0.3 0.8 2.0
## x[1,29] 0.0 0.2 1.2 -2.0 -0.5 0.1 0.9 2.1
## x[1,30] 0.0 0.1 1.0 -1.7 -0.6 0.0 0.4 2.4
## x[1,31] -0.3 0.1 1.0 -2.0 -0.8 -0.3 0.3 1.7
## x[1,32] -0.5 0.1 1.2 -3.3 -1.2 -0.7 0.3 1.6
## x[1,33] -0.7 0.2 1.2 -3.0 -1.3 -0.7 0.1 1.3
## x[1,34] -0.4 0.1 1.3 -2.3 -1.2 -0.5 0.2 2.3
## x[1,35] -0.6 0.2 1.4 -2.8 -1.3 -0.4 0.1 1.4
## x[1,36] -0.5 0.2 1.5 -3.4 -1.1 -0.2 0.5 1.8
## x[1,37] 0.1 0.2 1.3 -2.3 -0.7 0.2 0.8 2.5
## x[1,38] 0.0 0.2 1.6 -3.7 -0.8 -0.1 1.2 2.2
## x[1,39] 0.3 0.2 1.1 -1.7 -0.1 0.3 0.8 2.4
## x[1,40] 0.3 0.2 1.1 -1.9 -0.4 0.4 1.0 2.8
## x[1,41] 0.2 0.1 1.1 -2.2 -0.3 0.3 1.1 2.0
## x[1,42] -0.1 0.1 1.2 -2.2 -1.1 0.0 0.7 1.9
## x[1,43] -0.4 0.2 1.3 -2.2 -1.3 -0.2 0.1 2.9
## x[1,44] -0.4 0.2 1.3 -3.0 -0.8 -0.3 0.2 2.3
## x[1,45] -0.1 0.2 1.3 -2.5 -1.2 -0.1 0.7 2.5
## x[1,46] -0.3 0.2 1.3 -3.3 -0.9 -0.2 0.4 2.0
## x[1,47] -0.5 0.1 1.5 -3.1 -1.5 -0.4 0.4 2.5
## x[1,48] -0.3 0.2 1.3 -2.9 -0.9 -0.5 0.6 2.3
## x[1,49] 0.2 0.3 1.5 -2.7 -0.7 0.1 1.1 3.0
## x[1,50] -0.1 0.2 1.3 -2.3 -1.1 -0.1 0.8 1.8
## x[1,51] -0.1 0.3 1.3 -2.4 -1.0 -0.3 0.8 2.3
## x[1,52] -0.5 0.1 1.2 -3.5 -1.0 -0.3 0.0 1.1
## x[1,53] -0.4 0.2 1.2 -2.4 -1.5 -0.1 0.3 1.4
## x[1,54] -0.5 0.1 0.8 -2.1 -0.9 -0.5 0.0 0.9
## x[1,55] -0.5 0.2 1.1 -2.4 -1.3 -0.3 0.3 1.1
## x[1,56] -0.2 0.2 1.3 -2.5 -0.8 -0.2 0.7 1.6
## x[1,57] 0.3 0.2 1.3 -1.9 -0.6 0.2 1.3 2.0
## x[1,58] 0.1 0.2 1.2 -2.3 -0.5 0.1 1.0 1.9
## x[1,59] 0.3 0.3 1.5 -2.2 -0.5 0.5 1.4 2.8
## x[1,60] 0.2 0.3 1.2 -2.5 -0.5 0.3 1.0 2.4
## x[1,61] 0.1 0.3 1.1 -2.0 -0.5 0.0 0.5 2.4
## x[1,62] 0.4 0.2 1.2 -1.7 -0.3 0.3 1.1 2.4
## x[1,63] 0.4 NaN 1.5 -2.3 -0.5 0.3 1.2 3.0
## x[1,64] 0.6 0.1 1.0 -0.8 0.1 0.7 1.1 3.0
## x[1,65] 0.6 0.1 1.4 -1.7 -0.2 0.5 1.5 3.2
## x[1,66] 0.4 0.3 1.3 -1.6 -0.5 0.4 1.1 2.4
## x[1,67] 0.5 0.1 1.2 -1.5 -0.4 0.2 1.4 3.4
## x[1,68] 0.2 0.2 1.2 -1.6 -0.7 0.1 1.0 2.9
## x[1,69] 0.0 0.2 1.2 -2.4 -0.8 -0.1 0.6 2.2
## x[1,70] -0.4 0.1 0.9 -2.5 -1.0 -0.4 0.2 1.1
## x[1,71] -1.2 0.1 1.2 -3.3 -2.0 -1.0 -0.5 0.8
## x[1,72] -1.7 0.2 1.5 -4.8 -2.4 -1.6 -0.6 0.1
## x[1,73] -2.0 0.2 1.6 -5.7 -2.6 -1.7 -0.9 0.0
## x[1,74] -2.5 0.2 1.9 -7.3 -3.5 -2.1 -1.3 -0.1
## x[1,75] -2.9 0.2 2.1 -6.2 -4.3 -2.4 -1.3 -0.1
## x[1,76] -3.1 0.2 1.6 -6.7 -4.2 -3.2 -1.6 -0.5
## x[1,77] -2.9 0.2 1.3 -5.6 -3.7 -2.8 -2.0 -0.8
## x[1,78] -2.5 0.2 1.4 -4.8 -3.4 -2.6 -1.8 0.3
## x[1,79] -1.6 0.2 1.1 -3.6 -2.3 -1.6 -0.8 0.0
## x[1,80] -0.2 0.2 1.0 -1.6 -0.9 -0.2 0.4 1.8
## x[1,81] 2.6 0.2 1.6 0.5 1.6 2.3 3.5 6.0
## x[1,82] 3.5 0.2 1.7 0.7 2.4 3.5 4.8 7.0
## x[1,83] 3.8 0.2 1.9 1.3 2.2 3.2 5.5 7.1
## x[1,84] 3.5 0.2 1.8 0.9 2.1 3.3 4.7 6.8
## x[1,85] 1.6 0.2 1.1 -0.1 0.9 1.6 2.3 3.9
## x[1,86] 1.0 0.3 1.5 -1.1 0.1 0.7 1.6 4.7
## x[1,87] 0.8 0.2 1.5 -1.6 0.1 0.5 1.5 4.0
## x[1,88] 0.4 0.2 1.3 -1.4 -0.2 0.3 1.0 3.5
## x[1,89] 0.2 0.2 1.3 -2.1 -0.4 0.1 0.6 2.7
## x[1,90] 0.1 0.2 1.4 -3.3 -0.5 0.2 0.9 2.2
## x[1,91] -0.2 0.2 1.2 -2.9 -0.8 0.1 0.7 1.7
## x[1,92] -0.1 0.1 1.3 -2.3 -0.9 -0.1 0.7 2.1
## x[1,93] -0.2 0.1 1.0 -2.0 -0.9 -0.1 0.6 1.3
## x[1,94] 0.1 0.1 1.2 -2.2 -0.8 0.3 1.1 2.0
## x[1,95] 0.1 0.1 1.2 -1.6 -0.9 -0.2 1.1 2.2
## x[1,96] -0.6 0.1 1.1 -2.5 -1.3 -0.6 0.2 1.4
## x[1,97] -1.0 0.1 1.2 -3.1 -1.7 -1.2 -0.3 0.9
## x[1,98] -0.8 0.1 1.0 -2.5 -1.7 -0.5 -0.1 0.5
## x[1,99] -0.7 0.1 1.1 -2.7 -1.3 -0.6 0.1 1.3
## x[1,100] -0.5 0.2 1.3 -2.5 -1.3 -0.6 0.0 2.0
## x[1,101] -0.4 0.3 1.2 -2.2 -1.1 -0.7 0.2 2.1
## x[1,102] -0.4 0.1 1.0 -2.1 -1.2 -0.4 0.3 1.2
## x[1,103] -0.3 0.1 1.1 -2.2 -0.9 -0.4 0.4 1.4
## x[1,104] 0.0 0.1 1.1 -2.1 -0.8 0.1 0.7 2.2
## x[1,105] 0.9 0.2 1.7 -1.4 -0.4 0.6 1.7 5.0
## x[1,106] 1.2 0.2 1.5 -0.9 0.2 0.6 2.2 4.6
## x[1,107] 1.2 0.3 1.6 -0.8 0.1 1.0 2.0 5.6
## x[1,108] 0.8 0.2 1.0 -1.3 0.4 0.9 1.5 2.6
## x[1,109] 0.3 0.2 1.2 -1.7 -0.5 0.2 0.8 2.7
## x[1,110] -0.1 0.1 1.2 -1.7 -1.0 -0.2 0.5 2.3
## x[1,111] -0.6 0.2 1.1 -2.6 -1.3 -0.5 0.2 1.1
## x[1,112] -1.0 0.1 1.0 -3.3 -1.7 -0.9 -0.3 0.4
## x[1,113] -0.8 0.2 1.2 -2.7 -1.6 -0.8 -0.1 0.9
## x[1,114] -0.4 0.2 1.1 -2.8 -1.1 -0.2 0.3 1.4
## x[1,115] -0.3 0.1 1.2 -2.3 -1.0 -0.2 0.6 1.6
## x[1,116] -0.1 0.2 1.1 -2.4 -0.5 -0.1 0.4 2.0
## x[1,117] 0.2 0.2 1.4 -1.6 -0.9 0.1 0.9 2.6
## x[1,118] 0.6 0.2 1.3 -2.0 -0.2 0.5 1.2 3.4
## x[1,119] 0.7 0.2 1.3 -1.1 -0.3 0.6 1.4 2.8
## x[1,120] 0.5 0.1 0.9 -1.0 -0.2 0.4 1.2 2.1
## x[1,121] 0.2 0.2 1.0 -1.9 -0.4 0.3 0.9 1.6
## x[1,122] 0.2 0.1 1.0 -1.6 -0.4 0.2 0.8 1.8
## x[1,123] 0.1 0.2 1.3 -2.2 -0.6 0.5 1.0 1.7
## x[1,124] 0.4 0.2 1.2 -1.7 -0.3 0.4 1.3 2.6
## x[1,125] 0.3 0.2 1.2 -1.9 -0.6 0.1 0.9 3.1
## x[1,126] -0.5 0.1 0.9 -2.0 -0.9 -0.5 -0.1 1.6
## x[1,127] -0.9 0.2 1.2 -3.0 -1.6 -1.0 0.0 1.4
## x[1,128] -0.9 NaN 1.2 -2.9 -1.7 -0.8 0.0 1.3
## x[1,129] -0.9 0.1 0.9 -2.6 -1.5 -0.9 -0.3 0.7
## x[1,130] -0.5 0.2 1.3 -2.9 -1.3 -0.4 0.2 1.7
## x[1,131] -0.4 0.1 0.9 -2.0 -0.9 -0.4 0.1 1.1
## x[1,132] -0.4 0.1 1.4 -2.8 -1.1 -0.4 0.2 2.9
## x[1,133] -0.9 0.1 1.4 -3.3 -1.6 -0.9 -0.2 1.4
## x[1,134] -1.1 0.2 1.1 -3.0 -1.9 -1.1 -0.3 0.6
## x[1,135] -1.4 0.2 1.1 -3.8 -2.3 -1.1 -0.5 0.2
## x[1,136] -1.6 0.2 1.5 -4.2 -2.6 -1.6 -0.8 0.5
## x[1,137] -1.1 0.2 1.1 -3.0 -1.9 -1.2 -0.1 0.6
## x[1,138] -1.2 0.1 0.9 -2.9 -1.7 -1.0 -0.4 0.2
## x[1,139] -0.3 0.2 1.4 -3.1 -1.1 -0.2 0.2 2.5
## x[1,140] 0.3 0.2 1.0 -1.2 -0.3 0.1 0.6 2.2
## x[1,141] 1.4 0.2 1.3 -0.5 0.5 1.2 2.1 4.0
## x[1,142] 1.2 0.2 1.5 -1.4 0.2 1.1 2.1 3.7
## x[1,143] 0.3 0.2 1.6 -2.9 -0.8 0.4 0.9 3.1
## x[1,144] -0.9 0.2 1.4 -3.0 -1.6 -1.0 -0.3 2.2
## x[1,145] -2.9 0.3 1.7 -6.8 -3.8 -2.8 -1.6 -0.6
## x[1,146] -3.1 0.2 1.7 -6.6 -4.1 -2.6 -1.8 -0.8
## x[1,147] -1.2 0.2 1.1 -3.2 -2.0 -1.1 -0.5 0.6
## x[1,148] -0.2 0.2 1.2 -2.2 -1.1 -0.3 0.6 1.9
## x[1,149] 0.1 0.1 1.1 -1.6 -0.4 0.1 0.9 2.2
## x[1,150] 0.8 0.1 1.2 -1.1 -0.1 0.7 1.8 3.0
## x[1,151] 2.1 0.2 1.5 -0.1 1.0 2.0 3.1 4.7
## x[1,152] 3.6 0.3 2.0 1.0 2.1 3.5 4.7 8.1
## x[1,153] 3.5 0.3 1.8 1.2 2.1 3.2 4.4 7.2
## x[1,154] 1.9 0.3 1.4 0.2 0.8 1.6 2.6 4.8
## x[1,155] 0.8 0.1 1.0 -0.5 0.1 0.6 1.5 2.2
## x[1,156] 0.4 0.1 1.3 -1.4 -0.5 0.3 1.1 2.8
## x[1,157] 0.7 0.1 1.4 -1.5 -0.5 0.6 1.8 3.5
## x[1,158] 0.5 0.3 1.5 -2.8 -0.3 0.6 1.5 2.8
## x[1,159] 0.4 0.2 1.3 -1.8 -0.4 0.5 1.2 2.9
## x[1,160] 0.1 0.2 1.3 -2.6 -0.4 0.3 0.9 1.8
## x[1,161] -0.2 0.1 0.9 -2.4 -0.6 0.1 0.4 1.2
## x[1,162] -0.1 0.1 1.1 -2.0 -0.8 0.0 0.6 2.1
## x[1,163] -0.4 0.1 1.0 -2.1 -1.1 -0.3 0.1 1.4
## x[1,164] -0.4 0.1 1.4 -3.0 -1.0 -0.2 0.3 2.4
## x[1,165] -0.2 0.2 1.1 -2.0 -0.8 -0.2 0.4 1.8
## x[1,166] -0.2 0.1 1.5 -2.4 -1.1 -0.4 0.4 3.0
## x[1,167] -0.2 0.1 1.1 -1.7 -0.8 -0.4 0.4 2.4
## x[1,168] -0.5 0.1 1.2 -2.5 -1.4 -0.6 0.4 1.5
## x[1,169] -0.8 0.1 1.5 -3.2 -2.1 -0.7 0.1 1.9
## x[1,170] -1.0 0.2 1.5 -3.2 -2.1 -1.3 -0.2 1.8
## x[1,171] -1.2 0.1 1.4 -3.6 -2.4 -1.0 0.0 0.6
## x[1,172] -1.1 0.2 1.6 -4.5 -1.9 -0.9 0.2 1.3
## x[1,173] -0.3 0.1 1.1 -2.5 -0.6 -0.2 0.3 1.1
## x[1,174] 0.4 0.2 1.4 -2.0 -0.6 0.5 1.2 3.4
## x[1,175] 0.7 0.2 1.6 -2.5 -0.3 0.5 1.5 3.8
## x[1,176] 0.6 0.1 1.1 -1.1 0.0 0.6 1.2 3.0
## x[1,177] 1.0 0.1 1.1 -1.0 0.4 1.1 1.8 2.9
## x[1,178] 0.9 0.1 1.2 -0.9 0.3 0.7 1.3 3.3
## x[1,179] 0.3 0.1 1.1 -2.1 -0.2 0.4 0.9 2.2
## x[1,180] -0.5 0.2 1.3 -2.9 -1.3 -0.4 0.4 1.4
## x[1,181] -1.9 0.2 1.5 -5.1 -2.7 -1.4 -0.8 0.0
## x[1,182] -2.4 0.2 1.2 -4.6 -3.2 -2.4 -1.5 -0.2
## x[1,183] -2.5 0.2 1.2 -4.7 -3.0 -2.4 -1.6 -0.5
## x[1,184] -2.8 0.2 1.6 -6.3 -3.6 -2.5 -1.7 -0.5
## x[1,185] -2.6 0.3 1.8 -6.9 -3.4 -2.5 -1.2 -0.2
## x[1,186] -2.5 0.2 1.6 -6.5 -3.0 -2.5 -1.4 -0.2
## x[1,187] -1.7 0.2 1.6 -5.4 -2.3 -1.3 -0.8 0.6
## x[1,188] 0.0 0.1 1.5 -2.9 -1.0 0.0 0.6 2.5
## x[1,189] 1.5 0.1 1.4 -0.6 0.7 1.3 1.9 4.2
## x[1,190] 4.0 0.3 2.5 1.1 2.3 3.5 5.1 11.7
## x[1,191] 4.7 0.3 2.2 1.4 3.2 4.5 5.7 8.8
## x[1,192] 3.9 0.2 2.2 1.1 2.2 3.7 5.5 7.7
## x[1,193] 1.1 0.2 1.6 -1.9 0.1 0.7 2.1 4.5
## x[1,194] -0.1 0.2 1.4 -3.2 -1.0 0.2 0.8 2.1
## x[1,195] -0.6 0.2 1.5 -3.5 -1.5 -0.8 0.3 2.1
## x[1,196] -1.1 0.2 1.4 -3.8 -1.9 -1.2 -0.6 1.8
## x[1,197] -1.4 0.1 1.4 -4.2 -2.5 -1.4 -0.4 0.9
## x[1,198] -1.5 0.1 1.4 -3.9 -2.4 -1.4 -0.5 0.9
## x[1,199] -1.5 0.1 1.4 -4.3 -2.5 -1.4 -0.4 1.1
## x[1,200] -1.0 0.2 1.4 -4.2 -2.0 -0.6 0.0 1.1
## x[1,201] 0.7 0.1 1.2 -2.2 0.0 0.9 1.3 2.5
## x[1,202] 3.3 0.3 2.0 0.9 1.9 3.0 3.8 8.3
## x[1,203] 5.6 0.4 2.7 1.7 3.3 5.8 7.4 10.8
## x[1,204] 6.8 0.3 3.3 1.7 4.2 6.3 9.3 14.1
## x[1,205] 7.2 0.3 3.5 1.9 4.3 7.0 9.5 13.6
## x[1,206] 5.4 0.3 3.0 1.7 2.9 5.0 7.3 11.1
## x[1,207] 4.7 0.2 2.7 1.4 2.5 4.1 7.0 10.1
## x[1,208] 4.9 0.2 2.6 1.7 2.7 4.4 6.5 10.7
## x[1,209] 2.0 0.2 1.6 -0.3 1.0 1.9 3.1 5.4
## x[1,210] -0.1 0.1 1.2 -3.1 -0.8 0.1 0.6 1.4
## x[1,211] -1.7 0.2 1.5 -4.8 -2.3 -1.3 -0.9 0.4
## x[1,212] -3.0 0.3 2.1 -6.9 -4.4 -2.7 -1.3 -0.5
## x[1,213] -3.7 0.2 2.0 -7.0 -5.0 -3.4 -2.2 -1.3
## x[1,214] -3.7 0.3 1.9 -7.4 -5.1 -3.7 -2.5 -0.3
## x[1,215] -3.6 0.2 1.8 -7.3 -4.6 -3.5 -2.3 -0.6
## x[1,216] -2.3 0.2 1.9 -6.4 -3.3 -1.8 -1.0 0.2
## x[1,217] 0.4 0.2 1.2 -1.6 -0.2 0.3 0.8 3.4
## x[1,218] 1.3 0.2 1.3 -0.9 0.8 1.4 1.8 4.6
## x[1,219] 1.0 0.1 1.2 -0.9 0.1 1.0 1.9 3.1
## x[1,220] 0.4 0.2 1.3 -2.5 -0.1 0.4 1.0 2.8
## x[1,221] 0.0 0.2 1.2 -2.3 -0.6 0.1 0.6 2.0
## x[1,222] -0.8 0.1 1.3 -3.6 -1.7 -0.6 0.3 0.9
## x[1,223] -1.4 0.2 1.4 -4.7 -2.0 -1.0 -0.6 1.0
## x[1,224] -1.5 0.0 1.5 -4.5 -2.3 -1.1 -0.5 0.8
## x[1,225] -1.2 0.0 1.2 -3.4 -2.1 -1.0 -0.5 0.7
## x[1,226] -1.7 0.1 1.4 -4.8 -2.5 -1.3 -0.6 0.3
## x[1,227] -1.3 0.2 1.4 -4.4 -1.9 -1.2 -0.4 0.7
## x[1,228] -1.4 0.1 1.9 -5.5 -2.3 -1.1 -0.1 1.4
## x[1,229] -0.3 0.2 1.4 -3.3 -1.0 -0.1 0.5 2.0
## x[1,230] 0.1 0.2 1.2 -2.0 -0.4 0.0 0.6 2.2
## x[1,231] 0.1 0.1 1.4 -2.9 -0.8 0.1 1.2 2.2
## x[1,232] 0.2 0.1 1.3 -2.3 -0.6 0.4 1.1 2.1
## x[1,233] 0.4 0.2 1.3 -2.0 -0.4 0.6 1.3 2.5
## x[1,234] 0.2 0.1 1.2 -1.8 -0.4 0.2 0.7 2.8
## x[1,235] 0.3 0.2 1.5 -2.0 -0.5 0.4 1.4 2.9
## x[1,236] -0.1 0.1 1.1 -2.1 -0.9 0.0 0.7 1.8
## x[1,237] -0.3 0.2 1.2 -3.0 -0.8 -0.2 0.3 1.7
## x[1,238] -0.4 0.2 1.4 -3.2 -1.2 -0.5 0.7 2.3
## x[1,239] -0.5 0.2 1.2 -3.2 -1.0 -0.5 0.0 1.7
## x[1,240] -0.4 0.2 1.5 -2.8 -1.1 -0.4 0.1 2.3
## x[1,241] -0.2 0.2 1.4 -2.5 -1.2 0.0 0.4 2.2
## x[1,242] 0.2 0.3 1.5 -2.6 -0.6 0.1 1.0 3.2
## x[1,243] 0.1 0.2 1.3 -2.0 -0.6 0.0 0.6 3.1
## x[1,244] 0.0 0.2 1.3 -2.4 -0.7 -0.1 0.9 1.8
## x[1,245] -0.1 0.3 1.7 -3.3 -0.9 -0.3 0.9 2.3
## x[1,246] 0.0 0.2 1.6 -2.6 -0.9 -0.2 0.8 2.4
## x[1,247] -0.5 0.2 1.7 -4.1 -1.6 -0.3 0.5 2.1
## x[1,248] -0.2 0.2 1.6 -3.0 -1.0 -0.4 0.1 3.6
## x[1,249] -0.1 0.2 1.4 -2.4 -0.8 -0.3 0.6 3.6
## x[1,250] -0.3 0.2 1.1 -2.1 -1.0 -0.3 0.4 1.8
## x[1,251] -0.6 0.2 1.2 -2.5 -1.2 -0.6 0.0 1.5
## x[1,252] -0.7 0.3 1.4 -4.2 -1.0 -0.3 0.3 0.8
## x[1,253] -0.5 0.2 1.3 -3.5 -0.9 -0.3 0.2 1.5
## x[1,254] -0.3 0.2 1.1 -3.0 -0.9 -0.3 0.5 1.4
## x[1,255] 0.0 0.2 1.4 -2.9 -0.9 0.2 0.9 1.7
## x[1,256] -0.2 0.1 1.3 -2.8 -1.2 -0.3 0.7 2.2
## x[1,257] -0.2 0.2 1.1 -2.4 -0.7 -0.2 0.4 1.6
## x[1,258] -0.2 0.1 1.0 -1.9 -0.9 -0.1 0.5 1.7
## x[1,259] 0.0 0.2 1.5 -3.2 -1.0 0.1 1.1 2.8
## x[1,260] -0.1 0.2 1.4 -2.6 -1.0 -0.3 0.8 2.5
## x[1,261] -0.2 0.1 1.0 -2.1 -1.1 -0.2 0.3 1.8
## x[1,262] -0.5 0.1 1.1 -2.3 -1.3 -0.6 0.2 1.7
## x[1,263] -0.9 0.2 1.1 -2.6 -2.0 -0.6 -0.2 1.4
## x[1,264] -1.1 0.1 1.1 -2.8 -1.7 -1.3 -0.3 0.8
## x[1,265] -1.9 0.2 1.4 -4.9 -2.5 -1.5 -1.1 0.3
## x[1,266] -1.9 0.2 1.9 -7.0 -2.7 -1.6 -0.8 0.9
## x[1,267] -1.4 0.0 1.2 -3.4 -2.2 -1.5 -0.6 1.0
## x[1,268] -1.3 0.1 1.1 -3.5 -1.9 -1.2 -0.6 0.7
## x[1,269] -0.8 0.1 0.9 -2.2 -1.5 -0.8 -0.3 0.4
## x[1,270] -0.6 0.2 1.4 -3.0 -1.2 -0.5 0.2 2.3
## x[1,271] -0.4 0.2 1.2 -3.1 -1.1 -0.1 0.1 1.5
## x[1,272] 0.3 0.1 1.1 -1.9 -0.4 0.1 1.2 2.3
## x[1,273] 1.2 0.2 1.3 -0.5 0.3 1.0 1.9 3.5
## x[1,274] 1.9 0.2 1.3 0.2 0.9 1.9 2.5 5.2
## x[1,275] 2.5 0.2 1.5 0.1 1.3 2.3 3.8 5.3
## x[1,276] 2.7 0.2 1.5 0.4 1.4 2.6 3.7 5.6
## x[1,277] 1.8 0.2 1.3 -0.3 0.7 2.0 2.6 4.4
## x[1,278] 1.1 0.2 1.3 -0.6 0.1 0.8 1.9 4.3
## x[1,279] 0.7 0.1 1.4 -1.6 -0.1 0.4 1.5 3.9
## x[1,280] 0.7 0.2 1.2 -1.1 -0.1 0.7 1.4 3.0
## x[1,281] 0.7 0.2 1.6 -1.7 -0.4 0.3 1.8 4.5
## x[1,282] 0.1 0.2 1.5 -3.2 -0.4 0.1 0.7 2.4
## x[1,283] -0.3 0.2 1.3 -2.0 -1.2 -0.3 0.2 1.3
## x[1,284] -0.5 0.2 1.2 -2.7 -1.3 -0.6 0.2 1.7
## x[1,285] -1.1 0.2 1.2 -3.3 -1.8 -1.0 -0.4 1.0
## x[1,286] -1.3 0.1 1.4 -3.8 -2.3 -1.2 -0.4 1.1
## x[1,287] -1.3 0.3 1.5 -4.4 -1.8 -1.2 -0.4 1.0
## x[1,288] -1.0 0.2 1.1 -2.9 -1.5 -1.0 -0.2 1.0
## x[1,289] -1.4 0.2 1.2 -3.1 -2.2 -1.7 -0.7 1.1
## x[1,290] -1.2 0.1 1.4 -4.2 -2.0 -1.1 -0.1 0.9
## x[1,291] -0.7 0.1 1.2 -2.8 -1.8 -0.6 0.0 1.9
## x[1,292] 0.2 0.1 1.1 -1.5 -0.4 0.0 0.8 2.4
## x[1,293] 0.5 0.1 1.6 -2.1 -0.6 0.4 1.6 3.9
## x[1,294] 0.6 0.3 1.5 -1.7 -0.6 0.3 1.4 3.5
## x[1,295] 0.7 0.2 1.4 -1.6 -0.3 0.7 1.4 3.5
## x[1,296] 0.4 0.2 1.3 -2.4 -0.3 0.4 1.1 2.8
## x[1,297] 0.2 0.2 1.5 -1.8 -0.8 0.2 1.2 3.8
## x[1,298] 0.6 0.2 1.2 -1.3 -0.2 0.4 1.2 3.0
## x[1,299] 1.1 0.2 1.6 -1.9 0.2 0.9 2.3 4.4
## x[1,300] 1.9 0.1 1.3 -0.2 1.1 1.9 2.4 4.6
## x[1,301] 2.4 0.2 1.8 -0.4 1.0 2.5 3.5 5.5
## x[1,302] 2.8 0.2 1.8 0.2 1.4 2.4 4.0 6.1
## x[1,303] 2.9 0.2 1.7 0.3 1.6 3.0 3.7 5.7
## x[1,304] 1.3 0.2 1.4 -0.8 0.3 1.1 2.2 4.1
## x[1,305] -0.3 0.2 1.7 -3.7 -1.2 0.1 0.7 2.6
## x[1,306] -0.9 0.2 1.6 -4.4 -1.4 -0.6 0.1 1.2
## x[1,307] -1.2 0.2 2.0 -5.8 -2.2 -1.0 -0.1 2.1
## x[1,308] -1.2 NaN 1.8 -4.8 -2.6 -1.2 0.0 2.4
## x[1,309] -1.5 0.1 1.5 -4.7 -2.6 -1.1 -0.5 0.8
## x[1,310] -1.0 0.1 1.3 -3.4 -1.7 -1.0 0.0 1.6
## x[1,311] -0.2 0.1 1.0 -2.0 -1.0 -0.3 0.5 1.4
## x[1,312] 0.2 0.1 1.2 -2.2 -0.5 0.0 0.8 2.5
## x[1,313] 0.5 0.2 1.2 -1.6 -0.2 0.5 1.1 2.4
## x[1,314] 0.7 0.2 1.1 -1.0 -0.1 0.6 1.3 2.7
## x[1,315] 0.9 0.2 1.4 -1.6 -0.1 0.7 1.8 3.3
## x[1,316] 0.8 0.1 1.3 -1.0 0.2 0.6 1.1 3.9
## x[1,317] 0.5 0.1 1.6 -1.5 -0.6 -0.1 1.3 4.4
## x[1,318] 0.1 0.2 1.4 -2.4 -1.0 0.0 0.9 2.9
## x[1,319] 0.0 0.2 1.1 -1.9 -0.8 0.1 0.8 1.6
## x[1,320] -0.1 0.2 1.0 -2.2 -0.4 -0.1 0.6 1.2
## x[1,321] -0.2 0.1 1.2 -3.2 -0.9 0.1 0.6 1.5
## x[1,322] -0.9 0.2 1.5 -4.1 -1.6 -0.7 -0.2 1.9
## x[1,323] -1.2 0.1 1.3 -3.8 -1.8 -1.1 -0.5 1.5
## x[1,324] -0.9 0.2 1.7 -4.3 -1.9 -0.9 0.0 3.2
## x[1,325] -0.7 0.1 1.2 -2.5 -1.5 -1.1 -0.4 2.2
## x[1,326] -0.6 0.1 1.3 -2.5 -1.3 -0.7 0.4 2.1
## x[1,327] -0.5 0.2 1.2 -2.9 -1.4 -0.2 0.3 1.7
## x[1,328] -0.3 0.2 1.2 -2.5 -1.2 -0.1 0.7 1.4
## x[1,329] -0.3 0.2 1.2 -2.5 -1.0 -0.4 0.3 2.2
## x[1,330] -0.4 0.1 1.4 -3.0 -1.1 -0.3 0.3 2.4
## x[1,331] -0.5 0.1 1.6 -3.1 -1.8 -0.5 0.7 2.7
## x[1,332] -0.1 0.2 1.4 -2.0 -1.0 -0.5 0.9 2.9
## x[1,333] -0.5 0.2 1.0 -2.8 -1.2 -0.3 0.2 0.8
## x[1,334] -0.4 0.2 1.2 -2.7 -0.9 -0.5 0.4 1.4
## x[1,335] -0.3 0.2 1.2 -3.4 -0.8 -0.2 0.5 1.9
## x[1,336] -0.9 0.2 1.3 -4.1 -1.4 -0.9 -0.1 1.1
## x[1,337] -2.5 0.2 1.7 -6.1 -3.7 -1.8 -1.2 -0.3
## x[1,338] -3.0 0.2 1.8 -6.3 -4.2 -2.5 -1.7 -0.5
## x[1,339] -3.4 0.2 2.1 -7.6 -4.4 -3.3 -1.9 -0.5
## x[1,340] -2.8 0.1 1.4 -5.4 -3.7 -2.6 -1.7 -0.6
## x[1,341] -2.0 0.1 1.4 -4.2 -3.2 -1.8 -0.8 0.4
## x[1,342] -1.2 0.2 1.4 -4.2 -1.8 -1.1 -0.3 0.7
## x[1,343] 0.4 0.1 1.2 -1.9 -0.2 0.3 1.1 2.8
## x[1,344] 2.1 0.2 1.5 -0.2 0.9 2.0 3.3 4.8
## x[1,345] 3.5 0.2 2.0 0.6 1.6 3.4 4.6 7.0
## x[1,346] 3.7 0.1 1.4 1.2 2.7 3.5 4.7 6.3
## x[1,347] 3.9 0.2 2.0 0.8 2.4 4.2 5.1 7.6
## x[1,348] 3.6 0.2 2.0 0.8 2.2 3.5 4.7 7.8
## x[1,349] 3.0 0.3 1.8 0.4 2.0 2.8 4.1 6.4
## x[1,350] 2.2 0.1 1.7 -1.0 1.0 2.0 3.2 5.2
## x[1,351] 1.5 0.1 1.3 -1.2 0.7 1.5 2.2 3.8
## x[1,352] 0.9 0.1 1.6 -2.2 -0.2 1.1 1.9 4.5
## x[1,353] 0.2 0.0 1.4 -2.3 -0.7 0.4 0.8 3.5
## x[1,354] -0.4 0.2 1.2 -2.7 -1.1 -0.5 0.7 1.5
## x[1,355] -0.4 0.1 1.3 -2.4 -1.4 -0.6 0.3 1.9
## x[1,356] -0.3 0.2 1.0 -1.8 -1.2 -0.5 0.1 1.6
## x[1,357] -0.7 0.1 1.3 -2.9 -1.5 -0.7 0.3 1.4
## x[1,358] -0.9 0.2 1.3 -2.9 -2.1 -0.9 0.1 1.1
## x[1,359] -1.2 0.2 1.2 -3.4 -1.8 -0.9 -0.2 0.4
## x[1,360] -1.5 0.2 1.1 -3.2 -2.4 -1.3 -0.6 0.3
## x[1,361] -1.4 0.1 0.9 -3.0 -2.1 -1.2 -0.8 -0.2
## x[1,362] -1.7 0.1 1.2 -3.7 -2.7 -1.6 -0.6 0.2
## x[1,363] -1.7 NaN 1.3 -4.0 -2.6 -1.5 -0.8 0.4
## x[1,364] -1.1 0.2 1.5 -3.3 -2.2 -1.1 -0.2 1.5
## x[1,365] -0.6 0.2 1.7 -3.6 -1.8 -0.4 0.4 2.8
## x[1,366] -0.2 0.2 1.4 -2.1 -1.3 -0.3 0.7 2.5
## x[1,367] -0.2 0.1 1.3 -2.5 -1.4 0.0 0.5 2.0
## x[1,368] -0.2 NaN 1.2 -2.7 -1.0 0.0 0.7 1.5
## x[1,369] 0.3 0.1 0.9 -1.4 0.0 0.4 1.0 1.7
## x[1,370] 1.1 0.3 1.2 -1.8 0.5 1.1 1.8 3.4
## x[1,371] 1.7 0.1 1.3 -0.1 0.8 1.5 2.6 3.9
## x[1,372] 1.3 0.1 1.4 -1.5 0.6 1.3 1.9 3.7
## x[1,373] 0.7 0.1 1.1 -2.0 0.0 0.6 1.1 2.6
## x[1,374] 0.1 0.2 1.5 -2.3 -0.7 0.2 0.8 2.5
## x[1,375] 0.2 0.1 1.3 -2.1 -0.5 0.0 1.0 3.2
## x[1,376] -0.5 0.2 1.5 -3.5 -1.0 -0.5 0.2 1.5
## x[1,377] -1.1 0.3 1.6 -4.5 -1.5 -0.8 -0.3 0.7
## x[1,378] -1.2 0.2 1.6 -4.6 -1.8 -0.9 -0.3 0.6
## x[1,379] -0.9 0.1 1.2 -3.6 -1.3 -0.6 -0.2 0.7
## x[1,380] -0.5 0.1 1.2 -2.9 -1.3 -0.4 0.4 1.4
## x[1,381] 0.4 0.2 1.2 -1.5 -0.3 0.2 1.0 2.9
## x[1,382] 1.1 0.2 1.4 -1.0 -0.1 1.0 2.2 3.8
## x[1,383] 1.2 0.2 1.2 -1.0 0.5 1.1 2.1 3.4
## x[1,384] 0.9 0.1 1.2 -0.8 0.0 0.7 1.5 3.5
## x[1,385] 0.7 0.1 1.3 -1.6 -0.3 0.7 1.5 3.1
## x[1,386] 0.7 0.2 1.1 -1.3 0.0 0.6 1.5 2.9
## x[1,387] 0.1 0.1 1.1 -1.9 -0.4 0.0 1.0 2.2
## x[1,388] 0.0 0.1 1.2 -2.0 -0.9 0.0 0.5 2.0
## x[1,389] -0.3 0.2 1.4 -3.7 -0.8 -0.4 0.3 2.3
## x[1,390] -0.5 0.1 1.2 -2.6 -1.3 -0.4 0.1 1.7
## x[1,391] -0.3 0.1 1.5 -3.1 -1.2 -0.2 0.6 2.1
## x[1,392] 0.3 0.1 1.0 -1.1 -0.5 0.2 1.0 2.0
## x[1,393] 1.1 NaN 1.2 -0.8 0.0 1.1 1.9 3.6
## x[1,394] 1.1 0.2 1.1 -0.6 0.5 1.0 1.7 3.4
## x[1,395] 1.3 0.2 1.5 -0.9 -0.1 1.2 2.5 4.0
## x[1,396] 1.7 0.2 1.3 -0.2 0.6 1.5 2.6 4.5
## x[1,397] 3.2 0.3 1.8 0.8 2.2 2.8 4.2 7.1
## x[1,398] 3.9 0.3 2.1 0.8 2.3 3.7 5.7 8.4
## x[1,399] 3.2 0.3 2.2 0.2 1.3 2.8 4.8 7.8
## x[1,400] 2.1 0.3 1.4 0.1 0.9 2.2 2.9 4.5
## x[1,401] 0.3 0.3 1.0 -1.3 -0.4 0.4 0.9 2.3
## x[1,402] -0.9 0.2 1.6 -3.5 -1.9 -0.9 0.2 2.2
## x[1,403] -1.2 0.1 1.4 -3.7 -2.0 -0.8 -0.4 1.3
## x[1,404] -1.3 0.1 1.2 -3.9 -1.9 -1.3 -0.5 0.9
## x[1,405] -2.1 0.1 1.5 -4.9 -3.1 -1.9 -1.0 0.3
## x[1,406] -2.7 0.2 1.5 -6.1 -3.7 -2.7 -1.5 -0.6
## x[1,407] -3.3 0.3 1.8 -6.7 -3.8 -3.0 -1.9 -1.0
## x[1,408] -2.6 0.2 1.7 -5.8 -3.9 -2.1 -1.2 -0.4
## x[1,409] -1.5 0.2 1.4 -3.6 -2.9 -1.1 -0.5 0.8
## x[1,410] -0.1 0.1 1.0 -1.8 -0.6 -0.3 0.3 2.4
## x[1,411] 0.9 0.2 1.4 -1.6 0.0 0.7 2.1 3.3
## x[1,412] 2.0 0.2 1.7 -0.4 0.9 1.8 2.9 5.9
## x[1,413] 2.4 0.1 1.3 0.6 1.4 2.0 3.3 5.4
## x[1,414] 2.1 0.1 1.4 0.2 1.3 2.0 2.8 5.1
## x[1,415] 1.2 0.2 1.4 -1.0 0.2 1.3 2.0 3.9
## x[1,416] 0.5 0.2 1.0 -1.7 0.1 0.4 1.1 2.2
## x[1,417] -0.7 0.2 1.1 -2.4 -1.3 -0.8 0.1 1.8
## x[1,418] -2.3 0.2 1.8 -4.9 -4.0 -2.0 -0.7 0.4
## x[1,419] -3.1 0.2 1.9 -7.0 -4.3 -2.6 -1.6 -0.7
## x[1,420] -2.5 0.1 1.4 -5.7 -3.3 -2.0 -1.6 -0.7
## x[1,421] -0.7 0.2 1.2 -2.8 -1.4 -0.6 0.3 0.9
## x[1,422] 0.4 0.0 1.2 -2.1 -0.4 0.4 1.2 2.1
## x[1,423] 1.0 0.1 1.7 -1.8 -0.2 1.1 2.2 4.1
## x[1,424] 1.2 0.1 1.1 -0.4 0.6 1.2 2.0 3.2
## x[1,425] 1.3 0.2 1.1 0.1 0.5 1.2 1.7 3.6
## x[1,426] 1.0 0.1 1.0 -0.4 0.3 0.6 1.5 3.1
## x[1,427] 0.4 0.1 1.3 -1.2 -0.5 0.2 0.9 2.9
## x[1,428] -0.2 0.2 1.7 -3.7 -1.5 -0.1 0.8 2.6
## x[1,429] -0.8 0.1 1.2 -3.8 -1.3 -0.7 -0.1 1.9
## x[1,430] -1.0 0.2 1.8 -3.1 -2.3 -1.2 0.0 3.2
## x[1,431] -1.5 0.2 1.4 -3.6 -2.6 -1.5 -0.8 1.2
## x[1,432] -1.8 0.2 1.5 -5.0 -2.6 -1.5 -1.1 1.3
## x[1,433] -0.9 0.2 1.4 -3.5 -1.8 -0.9 0.0 1.6
## x[1,434] -0.4 0.2 1.0 -1.9 -0.9 -0.6 -0.2 2.0
## x[1,435] 0.0 0.1 1.2 -2.1 -0.7 -0.1 0.8 2.3
## x[1,436] 0.0 0.1 0.9 -1.2 -0.5 0.1 0.5 1.9
## x[1,437] -0.2 0.1 0.9 -2.1 -0.8 -0.1 0.3 1.3
## x[1,438] 0.1 0.1 1.3 -2.0 -0.7 -0.3 0.7 2.5
## x[1,439] 0.3 0.1 1.1 -1.8 -0.7 0.2 1.1 2.5
## x[1,440] 0.5 0.1 0.9 -0.9 -0.1 0.3 1.0 2.6
## x[1,441] 0.6 0.1 1.2 -1.6 -0.1 0.4 1.4 2.5
## x[1,442] 0.7 0.2 1.1 -0.6 0.0 0.4 1.3 3.3
## x[1,443] 1.0 0.1 1.6 -1.7 0.2 1.0 1.6 4.2
## x[1,444] 1.9 NaN 1.7 -0.4 1.0 1.6 2.9 5.5
## x[1,445] 3.1 0.1 1.7 0.5 1.7 3.0 4.4 6.1
## x[1,446] 3.6 0.2 2.3 0.8 1.8 3.0 4.9 8.0
## x[1,447] 3.3 0.2 1.7 1.1 2.0 3.2 4.3 7.1
## x[1,448] 2.6 0.2 1.7 0.7 1.6 2.3 3.1 6.0
## x[1,449] 1.5 0.2 1.4 -0.6 0.5 1.3 2.4 4.2
## x[1,450] 0.3 0.2 1.2 -1.3 -0.5 0.1 0.8 3.2
## x[1,451] -0.7 0.1 1.2 -2.6 -1.6 -0.6 0.0 1.7
## x[1,452] -1.6 0.1 1.4 -4.1 -2.2 -1.6 -1.0 0.8
## x[1,453] -3.0 0.2 1.9 -7.2 -3.9 -2.7 -1.6 -0.3
## x[1,454] -4.1 0.2 2.2 -8.4 -5.7 -4.0 -2.3 -0.6
## x[1,455] -4.4 0.2 1.8 -7.8 -5.6 -4.5 -3.3 -1.6
## x[1,456] -3.9 0.2 1.4 -6.8 -4.8 -4.0 -2.9 -1.3
## x[1,457] -3.8 0.3 1.7 -6.5 -4.8 -4.2 -2.6 -0.2
## x[1,458] -3.9 0.1 1.8 -8.3 -4.9 -3.4 -2.7 -1.8
## x[1,459] -4.0 0.2 2.1 -9.3 -4.7 -3.8 -2.5 -1.2
## x[1,460] -3.7 0.2 2.0 -8.7 -4.8 -3.1 -2.4 -1.3
## x[1,461] -3.8 0.2 1.7 -7.9 -4.8 -3.4 -2.4 -1.5
## x[1,462] -3.2 0.1 1.7 -6.9 -4.3 -3.2 -1.7 -1.0
## x[1,463] -2.4 0.2 1.2 -5.3 -2.8 -2.3 -1.6 -0.8
## x[1,464] -0.5 0.1 1.3 -2.5 -1.4 -0.5 0.3 2.0
## x[1,465] 1.6 0.1 1.1 -0.1 0.8 1.5 2.3 3.6
## x[1,466] 4.8 0.2 2.0 1.9 3.4 4.7 5.9 8.7
## x[1,467] 6.6 0.3 2.5 2.7 4.6 6.4 8.2 11.2
## x[1,468] 7.3 0.3 2.7 3.2 4.9 7.1 9.4 12.4
## x[1,469] 6.6 0.3 2.7 2.8 4.4 6.4 8.7 12.6
## x[1,470] 6.2 0.3 2.2 3.2 4.4 6.1 7.5 10.4
## x[1,471] 5.7 0.3 2.0 2.2 4.2 5.7 7.0 9.4
## x[1,472] 3.9 0.2 1.4 1.6 2.8 3.7 4.7 6.6
## x[1,473] 1.7 0.2 1.2 0.2 0.8 1.3 2.3 3.7
## x[1,474] 0.1 0.1 1.4 -2.3 -0.7 0.3 1.0 2.5
## x[1,475] -1.2 0.2 1.4 -4.4 -2.1 -0.9 -0.3 1.0
## x[1,476] -2.7 0.2 1.8 -6.3 -3.4 -2.5 -1.2 -0.3
## x[1,477] -3.3 0.2 1.7 -7.0 -4.4 -3.4 -1.9 -0.7
## x[1,478] -4.0 0.2 1.7 -6.8 -5.1 -4.0 -2.6 -0.8
## x[1,479] -4.4 0.2 2.0 -8.2 -5.7 -4.2 -2.7 -1.6
## x[1,480] -3.6 0.2 2.1 -8.2 -4.9 -3.1 -2.2 -0.8
## x[1,481] -1.9 0.1 1.1 -4.3 -2.6 -1.8 -1.3 0.1
## x[1,482] -1.2 0.1 1.3 -3.6 -2.1 -1.0 -0.4 0.9
## x[1,483] -1.1 0.1 1.2 -3.2 -1.8 -1.3 -0.1 0.9
## x[1,484] -1.2 0.2 1.3 -3.7 -2.1 -1.4 -0.7 1.3
## x[1,485] -0.9 0.1 1.2 -3.6 -1.6 -0.6 -0.1 1.0
## x[1,486] -0.3 0.1 1.1 -2.2 -0.9 -0.1 0.4 2.0
## x[1,487] -0.1 0.1 1.0 -1.7 -0.8 -0.1 0.6 2.0
## x[1,488] 0.1 0.2 1.1 -1.8 -0.5 0.4 0.9 1.6
## x[1,489] 0.5 0.1 1.0 -1.3 -0.1 0.5 1.1 2.8
## x[1,490] 0.9 0.2 1.3 -0.9 0.0 0.6 1.4 4.2
## x[1,491] 1.2 0.2 1.5 -0.8 0.2 0.9 1.7 5.2
## x[1,492] 1.4 0.2 1.6 -0.7 0.3 1.2 1.9 5.6
## Z[1,1] 0.3 0.0 0.2 0.2 0.2 0.3 0.3 0.7
## Z[2,1] 0.2 0.0 0.1 0.1 0.2 0.2 0.3 0.5
## Z[3,1] 0.2 0.0 0.1 0.1 0.2 0.2 0.3 0.4
## sigma[1] 0.9 0.0 0.0 0.9 0.9 0.9 0.9 0.9
## log_lik[1] -0.9 0.0 0.2 -1.4 -1.0 -0.9 -0.8 -0.8
## log_lik[2] -0.8 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[3] -0.8 0.0 0.1 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[4] -1.0 0.0 0.2 -1.4 -1.1 -0.9 -0.8 -0.8
## log_lik[5] -0.9 0.0 0.1 -1.2 -0.9 -0.8 -0.8 -0.8
## log_lik[6] -0.8 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[7] -1.0 0.0 0.2 -1.4 -1.1 -0.9 -0.8 -0.8
## log_lik[8] -0.8 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[9] -0.8 0.0 0.1 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[10] -1.0 0.0 0.2 -1.3 -1.0 -0.9 -0.9 -0.8
## log_lik[11] -0.8 0.0 0.1 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[12] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[13] -1.1 0.0 0.2 -1.6 -1.2 -1.1 -1.0 -0.8
## log_lik[14] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[15] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[16] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[17] -0.8 0.0 0.0 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[18] -0.8 0.0 0.0 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[19] -1.0 0.0 0.2 -1.4 -1.1 -0.9 -0.9 -0.8
## log_lik[20] -0.9 0.0 0.1 -1.2 -0.9 -0.8 -0.8 -0.8
## log_lik[21] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[22] -1.1 0.0 0.3 -1.6 -1.2 -1.0 -0.9 -0.8
## log_lik[23] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[24] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[25] -1.0 0.0 0.2 -1.5 -1.1 -0.9 -0.8 -0.8
## log_lik[26] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[27] -0.8 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[28] -0.9 0.0 0.1 -1.2 -0.9 -0.8 -0.8 -0.8
## log_lik[29] -0.9 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[30] -0.8 0.0 0.0 -0.9 -0.9 -0.8 -0.8 -0.8
## log_lik[31] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[32] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[33] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[34] -0.9 0.0 0.2 -1.3 -0.9 -0.8 -0.8 -0.8
## log_lik[35] -0.8 0.0 0.1 -1.1 -0.8 -0.8 -0.8 -0.8
## log_lik[36] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[37] -0.9 0.0 0.2 -1.6 -1.0 -0.8 -0.8 -0.8
## log_lik[38] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[39] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[40] -0.9 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[41] -0.8 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[42] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[43] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[44] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[45] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[46] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[47] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[48] -0.8 0.0 0.0 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[49] -0.8 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[50] -0.8 0.0 0.1 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[51] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[52] -1.0 0.0 0.2 -1.6 -1.1 -1.0 -0.8 -0.8
## log_lik[53] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[54] -0.8 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[55] -1.0 0.0 0.2 -1.7 -1.1 -0.9 -0.8 -0.8
## log_lik[56] -0.9 0.0 0.1 -1.2 -0.9 -0.8 -0.8 -0.8
## log_lik[57] -0.9 0.0 0.1 -1.1 -0.9 -0.8 -0.8 -0.8
## log_lik[58] -0.8 0.0 0.1 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[59] -0.8 0.0 0.0 -1.0 -0.8 -0.8 -0.8 -0.8
## log_lik[60] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[61] -0.8 0.0 0.0 -0.9 -0.9 -0.8 -0.8 -0.8
## log_lik[62] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[63] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[64] -0.9 0.0 0.1 -1.2 -0.9 -0.8 -0.8 -0.8
## log_lik[65] -0.8 0.0 0.1 -1.0 -0.9 -0.8 -0.8 -0.8
## log_lik[66] -0.8 0.0 0.0 -0.9 -0.8 -0.8 -0.8 -0.8
## log_lik[67] -1.0 0.0 0.3 -1.4 -1.1 -0.9 -0.8 -0.8
## log_lik[68] -0.9 0.0 0.2 -1.4 -0.9 -0.8 -0.8 -0.8
## log_lik[69] -0.9 0.0 0.2 -1.2 -0.9 -0.8 -0.8 -0.8
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## psi[1] 2.0 0.1 1.3 0.3 1.1 1.9 2.6 4.8
## lp__ -2159.2 6.3 21.4 -2195.3 -2174.2 -2158.7 -2150.5 -2108.9
## n_eff Rhat
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## x[1,96] 82 1.0
## x[1,97] 70 1.0
## x[1,98] 75 1.0
## x[1,99] 365 1.0
## x[1,100] 60 1.0
## x[1,101] 19 1.1
## x[1,102] 59 1.0
## x[1,103] 137 1.0
## x[1,104] 92 1.0
## x[1,105] 96 1.0
## x[1,106] 51 1.0
## x[1,107] 30 1.0
## x[1,108] 36 1.1
## x[1,109] 62 1.0
## x[1,110] 90 1.0
## x[1,111] 29 1.1
## x[1,112] 63 1.0
## x[1,113] 43 1.0
## x[1,114] 40 1.0
## x[1,115] 70 1.0
## x[1,116] 43 1.0
## x[1,117] 64 1.0
## x[1,118] 48 1.0
## x[1,119] 56 1.0
## x[1,120] 43 1.0
## x[1,121] 40 1.0
## x[1,122] 52 1.0
## x[1,123] 54 1.0
## x[1,124] 35 1.0
## x[1,125] 54 1.0
## x[1,126] 53 1.0
## x[1,127] 65 1.0
## x[1,128] -1763 1.0
## x[1,129] 56 1.0
## x[1,130] 37 1.0
## x[1,131] 105 1.0
## x[1,132] 174 1.0
## x[1,133] 92 1.0
## x[1,134] 44 1.0
## x[1,135] 35 1.0
## x[1,136] 43 1.0
## x[1,137] 48 1.0
## x[1,138] 51 1.0
## x[1,139] 69 1.0
## x[1,140] 34 1.1
## x[1,141] 49 1.0
## x[1,142] 63 1.0
## x[1,143] 84 1.0
## x[1,144] 62 1.0
## x[1,145] 39 1.0
## x[1,146] 50 1.0
## x[1,147] 53 1.0
## x[1,148] 55 1.0
## x[1,149] 138 1.0
## x[1,150] 78 1.0
## x[1,151] 45 1.0
## x[1,152] 43 1.0
## x[1,153] 50 1.0
## x[1,154] 28 1.0
## x[1,155] 51 1.0
## x[1,156] 90 1.0
## x[1,157] 107 1.0
## x[1,158] 36 1.0
## x[1,159] 63 1.0
## x[1,160] 42 1.0
## x[1,161] 64 1.0
## x[1,162] 120 1.0
## x[1,163] 68 1.0
## x[1,164] 157 1.0
## x[1,165] 32 1.0
## x[1,166] 106 1.0
## x[1,167] 68 1.0
## x[1,168] 110 1.0
## x[1,169] 267 1.0
## x[1,170] 66 1.0
## x[1,171] 155 1.0
## x[1,172] 62 1.0
## x[1,173] 86 1.0
## x[1,174] 61 1.0
## x[1,175] 67 1.0
## x[1,176] 88 1.0
## x[1,177] 92 1.0
## x[1,178] 82 1.0
## x[1,179] 62 1.0
## x[1,180] 50 1.0
## x[1,181] 69 1.0
## x[1,182] 49 1.0
## x[1,183] 35 1.0
## x[1,184] 42 1.0
## x[1,185] 38 1.0
## x[1,186] 57 1.0
## x[1,187] 54 1.0
## x[1,188] 107 1.0
## x[1,189] 91 1.0
## x[1,190] 59 1.0
## x[1,191] 59 1.0
## x[1,192] 91 1.0
## x[1,193] 84 1.0
## x[1,194] 87 1.0
## x[1,195] 93 1.0
## x[1,196] 58 1.0
## x[1,197] 214 1.0
## x[1,198] 131 1.0
## x[1,199] 168 1.0
## x[1,200] 74 1.0
## x[1,201] 81 1.0
## x[1,202] 50 1.0
## x[1,203] 52 1.0
## x[1,204] 103 1.0
## x[1,205] 115 1.0
## x[1,206] 77 1.0
## x[1,207] 149 1.0
## x[1,208] 112 1.0
## x[1,209] 101 1.0
## x[1,210] 86 1.0
## x[1,211] 71 1.0
## x[1,212] 51 1.0
## x[1,213] 102 1.0
## x[1,214] 44 1.0
## x[1,215] 61 1.0
## x[1,216] 62 1.0
## x[1,217] 62 1.0
## x[1,218] 34 1.0
## x[1,219] 72 1.0
## x[1,220] 46 1.0
## x[1,221] 44 1.0
## x[1,222] 106 1.0
## x[1,223] 77 1.0
## x[1,224] 1036 1.0
## x[1,225] 12074 1.0
## x[1,226] 119 1.0
## x[1,227] 73 1.0
## x[1,228] 203 1.0
## x[1,229] 43 1.0
## x[1,230] 43 1.0
## x[1,231] 199 1.0
## x[1,232] 175 1.0
## x[1,233] 41 1.0
## x[1,234] 92 1.0
## x[1,235] 91 1.0
## x[1,236] 62 1.0
## x[1,237] 38 1.0
## x[1,238] 37 1.0
## x[1,239] 24 1.0
## x[1,240] 55 1.0
## x[1,241] 56 1.0
## x[1,242] 32 1.0
## x[1,243] 70 1.0
## x[1,244] 43 1.1
## x[1,245] 38 1.0
## x[1,246] 46 1.0
## x[1,247] 87 1.0
## x[1,248] 55 1.0
## x[1,249] 40 1.0
## x[1,250] 39 1.0
## x[1,251] 34 1.0
## x[1,252] 21 1.1
## x[1,253] 63 1.0
## x[1,254] 41 1.0
## x[1,255] 44 1.0
## x[1,256] 313 1.0
## x[1,257] 49 1.0
## x[1,258] 54 1.0
## x[1,259] 63 1.0
## x[1,260] 72 1.0
## x[1,261] 131 1.0
## x[1,262] 95 1.0
## x[1,263] 34 1.0
## x[1,264] 60 1.0
## x[1,265] 79 1.0
## x[1,266] 69 1.0
## x[1,267] 1998 1.0
## x[1,268] 156 1.0
## x[1,269] 53 1.0
## x[1,270] 38 1.0
## x[1,271] 41 1.0
## x[1,272] 62 1.0
## x[1,273] 47 1.0
## x[1,274] 42 1.0
## x[1,275] 49 1.0
## x[1,276] 42 1.0
## x[1,277] 45 1.0
## x[1,278] 57 1.0
## x[1,279] 97 1.0
## x[1,280] 31 1.0
## x[1,281] 81 1.0
## x[1,282] 47 1.0
## x[1,283] 30 1.0
## x[1,284] 33 1.0
## x[1,285] 63 1.0
## x[1,286] 136 1.0
## x[1,287] 30 1.0
## x[1,288] 23 1.0
## x[1,289] 49 1.0
## x[1,290] 102 1.0
## x[1,291] 83 1.0
## x[1,292] 86 1.0
## x[1,293] 128 1.0
## x[1,294] 33 1.0
## x[1,295] 34 1.0
## x[1,296] 27 1.0
## x[1,297] 39 1.0
## x[1,298] 59 1.0
## x[1,299] 85 1.0
## x[1,300] 84 1.0
## x[1,301] 104 1.0
## x[1,302] 141 1.0
## x[1,303] 109 1.0
## x[1,304] 83 1.0
## x[1,305] 115 1.0
## x[1,306] 101 1.0
## x[1,307] 93 1.0
## x[1,308] -136 1.0
## x[1,309] 153 1.0
## x[1,310] 331 1.0
## x[1,311] 49 1.0
## x[1,312] 69 1.0
## x[1,313] 66 1.0
## x[1,314] 36 1.0
## x[1,315] 62 1.0
## x[1,316] 109 1.0
## x[1,317] 163 1.0
## x[1,318] 54 1.0
## x[1,319] 42 1.0
## x[1,320] 41 1.0
## x[1,321] 63 1.0
## x[1,322] 65 1.0
## x[1,323] 100 1.0
## x[1,324] 59 1.0
## x[1,325] 72 1.0
## x[1,326] 83 1.0
## x[1,327] 61 1.0
## x[1,328] 48 1.0
## x[1,329] 63 1.0
## x[1,330] 167 1.0
## x[1,331] 213 1.0
## x[1,332] 66 1.0
## x[1,333] 29 1.0
## x[1,334] 48 1.0
## x[1,335] 67 1.0
## x[1,336] 53 1.0
## x[1,337] 66 1.0
## x[1,338] 134 1.0
## x[1,339] 93 1.0
## x[1,340] 179 1.0
## x[1,341] 320 1.0
## x[1,342] 44 1.1
## x[1,343] 63 1.0
## x[1,344] 64 1.0
## x[1,345] 87 1.0
## x[1,346] 101 1.0
## x[1,347] 97 1.0
## x[1,348] 85 1.0
## x[1,349] 49 1.0
## x[1,350] 134 1.0
## x[1,351] 101 1.0
## x[1,352] 132 1.0
## x[1,353] 8750 1.0
## x[1,354] 53 1.0
## x[1,355] 92 1.0
## x[1,356] 33 1.0
## x[1,357] 87 1.0
## x[1,358] 73 1.0
## x[1,359] 52 1.0
## x[1,360] 43 1.0
## x[1,361] 45 1.0
## x[1,362] 157 1.0
## x[1,363] -312 1.0
## x[1,364] 74 1.0
## x[1,365] 114 1.0
## x[1,366] 40 1.0
## x[1,367] 102 1.0
## x[1,368] -415 1.0
## x[1,369] 47 1.1
## x[1,370] 22 1.1
## x[1,371] 114 1.0
## x[1,372] 105 1.0
## x[1,373] 61 1.0
## x[1,374] 54 1.0
## x[1,375] 153 1.0
## x[1,376] 41 1.0
## x[1,377] 38 1.0
## x[1,378] 96 1.0
## x[1,379] 130 1.0
## x[1,380] 143 1.0
## x[1,381] 60 1.0
## x[1,382] 42 1.0
## x[1,383] 39 1.0
## x[1,384] 77 1.0
## x[1,385] 234 1.0
## x[1,386] 55 1.0
## x[1,387] 468 1.0
## x[1,388] 126 1.0
## x[1,389] 54 1.0
## x[1,390] 84 1.0
## x[1,391] 110 1.0
## x[1,392] 259 1.0
## x[1,393] -337 1.0
## x[1,394] 44 1.0
## x[1,395] 69 1.0
## x[1,396] 47 1.0
## x[1,397] 32 1.0
## x[1,398] 49 1.0
## x[1,399] 51 1.0
## x[1,400] 30 1.1
## x[1,401] 12 1.2
## x[1,402] 41 1.0
## x[1,403] 94 1.0
## x[1,404] 90 1.0
## x[1,405] 174 1.0
## x[1,406] 48 1.0
## x[1,407] 36 1.0
## x[1,408] 54 1.0
## x[1,409] 53 1.0
## x[1,410] 50 1.0
## x[1,411] 67 1.0
## x[1,412] 84 1.0
## x[1,413] 90 1.0
## x[1,414] 155 1.0
## x[1,415] 77 1.0
## x[1,416] 43 1.0
## x[1,417] 55 1.0
## x[1,418] 74 1.0
## x[1,419] 69 1.0
## x[1,420] 90 1.0
## x[1,421] 58 1.0
## x[1,422] 648 1.0
## x[1,423] 251 1.0
## x[1,424] 94 1.0
## x[1,425] 36 1.0
## x[1,426] 65 1.0
## x[1,427] 116 1.0
## x[1,428] 86 1.0
## x[1,429] 159 1.0
## x[1,430] 124 1.0
## x[1,431] 83 1.0
## x[1,432] 50 1.0
## x[1,433] 37 1.0
## x[1,434] 43 1.1
## x[1,435] 69 1.0
## x[1,436] 133 1.0
## x[1,437] 43 1.1
## x[1,438] 94 1.0
## x[1,439] 75 1.0
## x[1,440] 61 1.0
## x[1,441] 79 1.0
## x[1,442] 46 1.0
## x[1,443] 130 1.0
## x[1,444] -477 1.0
## x[1,445] 138 1.0
## x[1,446] 125 1.0
## x[1,447] 54 1.0
## x[1,448] 100 1.0
## x[1,449] 62 1.0
## x[1,450] 53 1.0
## x[1,451] 96 1.0
## x[1,452] 91 1.0
## x[1,453] 145 1.0
## x[1,454] 200 1.0
## x[1,455] 91 1.0
## x[1,456] 53 1.0
## x[1,457] 43 1.0
## x[1,458] 141 1.0
## x[1,459] 196 1.0
## x[1,460] 66 1.0
## x[1,461] 66 1.0
## x[1,462] 177 1.0
## x[1,463] 46 1.0
## x[1,464] 106 1.0
## x[1,465] 62 1.0
## x[1,466] 73 1.0
## x[1,467] 98 1.0
## x[1,468] 87 1.0
## x[1,469] 99 1.0
## x[1,470] 73 1.0
## x[1,471] 51 1.0
## x[1,472] 47 1.0
## x[1,473] 42 1.0
## x[1,474] 375 1.0
## x[1,475] 77 1.0
## x[1,476] 70 1.0
## x[1,477] 67 1.0
## x[1,478] 53 1.0
## x[1,479] 79 1.0
## x[1,480] 78 1.0
## x[1,481] 210 1.0
## x[1,482] 81 1.0
## x[1,483] 93 1.0
## x[1,484] 34 1.0
## x[1,485] 506 1.0
## x[1,486] 65 1.0
## x[1,487] 81 1.0
## x[1,488] 38 1.0
## x[1,489] 50 1.0
## x[1,490] 38 1.0
## x[1,491] 49 1.0
## x[1,492] 53 1.0
## Z[1,1] 44 1.0
## Z[2,1] 61 1.0
## Z[3,1] 56 1.0
## sigma[1] 42 1.0
## log_lik[1] 108 1.0
## log_lik[2] 18 1.0
## log_lik[3] 52 1.0
## log_lik[4] 44 1.0
## log_lik[5] 38 1.0
## log_lik[6] 38 1.0
## log_lik[7] 50 1.0
## log_lik[8] 48 1.0
## log_lik[9] 28 1.0
## log_lik[10] 84 1.0
## log_lik[11] 29 1.0
## log_lik[12] 34 1.0
## log_lik[13] 54 1.0
## log_lik[14] 38 1.0
## log_lik[15] 47 1.0
## log_lik[16] 33 1.0
## log_lik[17] 54 1.0
## log_lik[18] 39 1.0
## log_lik[19] 83 1.0
## log_lik[20] 37 1.0
## log_lik[21] 38 1.0
## log_lik[22] 35 1.0
## log_lik[23] 30 1.0
## log_lik[24] 31 1.0
## log_lik[25] 212 1.0
## log_lik[26] 33 1.0
## log_lik[27] 46 1.0
## log_lik[28] 214 1.0
## log_lik[29] 69 1.0
## log_lik[30] 93 1.0
## log_lik[31] 50 1.0
## log_lik[32] 35 1.0
## log_lik[33] 64 1.0
## log_lik[34] 49 1.0
## log_lik[35] 45 1.0
## log_lik[36] 55 1.0
## log_lik[37] 47 1.0
## log_lik[38] 18 1.0
## log_lik[39] 25 1.0
## log_lik[40] 56 1.0
## log_lik[41] 45 1.0
## log_lik[42] 54 1.0
## log_lik[43] 19 1.0
## log_lik[44] 17 1.0
## log_lik[45] 30 1.0
## log_lik[46] 32 1.0
## log_lik[47] 33 1.0
## log_lik[48] 41 1.0
## log_lik[49] 38 1.0
## log_lik[50] 36 1.0
## log_lik[51] 43 1.0
## log_lik[52] 53 1.0
## log_lik[53] 25 1.0
## log_lik[54] 37 1.0
## log_lik[55] 47 1.0
## log_lik[56] 27 1.0
## log_lik[57] 31 1.0
## log_lik[58] 21 1.0
## log_lik[59] 24 1.0
## log_lik[60] 24 1.0
## log_lik[61] 31 1.0
## log_lik[62] 25 1.0
## log_lik[63] 51 1.0
## log_lik[64] 48 1.0
## log_lik[65] 17 1.0
## log_lik[66] 29 1.0
## log_lik[67] 33 1.0
## log_lik[68] 16 1.0
## log_lik[69] 21 1.0
## log_lik[70] 54 1.0
## log_lik[71] 33 1.0
## log_lik[72] 36 1.0
## log_lik[73] 61 1.1
## log_lik[74] 57 1.0
## log_lik[75] 55 1.0
## log_lik[76] 96 1.0
## log_lik[77] 61 1.0
## log_lik[78] 53 1.0
## log_lik[79] 75 1.0
## log_lik[80] 166 1.0
## log_lik[81] 89 1.0
## log_lik[82] 69 1.0
## log_lik[83] 99 1.0
## log_lik[84] 72 1.0
## log_lik[85] 41 1.0
## log_lik[86] 40 1.0
## log_lik[87] 48 1.0
## log_lik[88] 21 1.0
## log_lik[89] 30 1.0
## log_lik[90] 37 1.0
## log_lik[91] 42 1.0
## log_lik[92] 27 1.0
## log_lik[93] 35 1.0
## log_lik[94] 37 1.0
## log_lik[95] 34 1.0
## log_lik[96] 50 1.0
## log_lik[97] 43 1.0
## log_lik[98] 48 1.0
## log_lik[99] 35 1.0
## log_lik[100] 73 1.0
## log_lik[101] 48 1.0
## log_lik[102] 57 1.0
## log_lik[103] 59 1.0
## log_lik[104] 50 1.0
## log_lik[105] 45 1.0
## log_lik[106] 18 1.0
## log_lik[107] 21 1.1
## log_lik[108] 32 1.1
## log_lik[109] 77 1.0
## log_lik[110] 30 1.0
## log_lik[111] 31 1.0
## log_lik[112] 14 1.0
## log_lik[113] 27 1.1
## log_lik[114] 23 1.1
## log_lik[115] 36 1.0
## log_lik[116] 35 1.0
## log_lik[117] 89 1.0
## log_lik[118] 45 1.0
## log_lik[119] 44 1.0
## log_lik[120] 59 1.0
## log_lik[121] 33 1.0
## log_lik[122] 45 1.0
## log_lik[123] 37 1.0
## log_lik[124] 55 1.0
## log_lik[125] 80 1.0
## log_lik[126] 58 1.0
## log_lik[127] 31 1.0
## log_lik[128] 24 1.0
## log_lik[129] 31 1.0
## log_lik[130] 25 1.0
## log_lik[131] 27 1.0
## log_lik[132] 35 1.0
## log_lik[133] 30 1.1
## log_lik[134] 29 1.1
## log_lik[135] 17 1.2
## log_lik[136] 30 1.1
## log_lik[137] 61 1.0
## log_lik[138] 71 1.0
## log_lik[139] 13 1.1
## log_lik[140] 72 1.0
## log_lik[141] 18 1.0
## log_lik[142] 24 1.0
## log_lik[143] 51 1.0
## log_lik[144] 23 1.1
## log_lik[145] 32 1.0
## log_lik[146] 25 1.0
## log_lik[147] 25 1.0
## log_lik[148] 44 1.0
## log_lik[149] 37 1.0
## log_lik[150] 37 1.0
## log_lik[151] 39 1.0
## log_lik[152] 40 1.0
## log_lik[153] 36 1.0
## log_lik[154] 19 1.1
## log_lik[155] 28 1.0
## log_lik[156] 31 1.0
## log_lik[157] 35 1.0
## log_lik[158] 66 1.0
## log_lik[159] 44 1.0
## log_lik[160] 30 1.0
## log_lik[161] 46 1.0
## log_lik[162] 41 1.0
## log_lik[163] 32 1.0
## log_lik[164] 19 1.1
## log_lik[165] 40 1.1
## log_lik[166] 37 1.1
## log_lik[167] 28 1.1
## log_lik[168] 49 1.1
## log_lik[169] 63 1.0
## log_lik[170] 82 1.0
## log_lik[171] 38 1.0
## log_lik[172] 41 1.0
## log_lik[173] 41 1.0
## log_lik[174] 26 1.0
## log_lik[175] 61 1.0
## log_lik[176] 29 1.0
## log_lik[177] 32 1.0
## log_lik[178] 40 1.0
## log_lik[179] 27 1.0
## log_lik[180] 56 1.0
## log_lik[181] 19 1.0
## log_lik[182] 45 1.1
## log_lik[183] 51 1.1
## log_lik[184] 52 1.0
## log_lik[185] 36 1.0
## log_lik[186] 46 1.0
## log_lik[187] 61 1.0
## log_lik[188] 31 1.1
## log_lik[189] 28 1.1
## log_lik[190] 55 1.0
## log_lik[191] 45 1.0
## log_lik[192] 46 1.0
## log_lik[193] 70 1.0
## log_lik[194] 57 1.0
## log_lik[195] 41 1.0
## log_lik[196] 32 1.0
## log_lik[197] 55 1.0
## log_lik[198] 59 1.0
## log_lik[199] 33 1.0
## log_lik[200] 40 1.0
## log_lik[201] 25 1.0
## log_lik[202] 15 1.2
## log_lik[203] 39 1.0
## log_lik[204] 39 1.0
## log_lik[205] 52 1.0
## log_lik[206] 33 1.0
## log_lik[207] 37 1.0
## log_lik[208] 21 1.1
## log_lik[209] 38 1.0
## log_lik[210] 32 1.1
## log_lik[211] 86 1.0
## log_lik[212] 37 1.0
## log_lik[213] 66 1.1
## log_lik[214] 35 1.0
## log_lik[215] 58 1.0
## log_lik[216] 58 1.0
## log_lik[217] 56 1.0
## log_lik[218] 61 1.0
## log_lik[219] 70 1.0
## log_lik[220] 74 1.0
## log_lik[221] 127 1.0
## log_lik[222] 45 1.0
## log_lik[223] -1160 1.0
## log_lik[224] -2394 1.0
## log_lik[225] 123 1.0
## log_lik[226] 116 1.0
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## log_lik[1441] 133 1.0
## log_lik[1442] 440 1.0
## log_lik[1443] 59 1.0
## log_lik[1444] 50 1.0
## log_lik[1445] 63 1.0
## log_lik[1446] 42 1.0
## log_lik[1447] 51 1.0
## log_lik[1448] 18 1.0
## log_lik[1449] 57 1.0
## log_lik[1450] 34 1.0
## log_lik[1451] 32 1.0
## log_lik[1452] 67 1.0
## log_lik[1453] 44 1.0
## log_lik[1454] 47 1.0
## log_lik[1455] 62 1.0
## log_lik[1456] 30 1.0
## log_lik[1457] 31 1.0
## log_lik[1458] 38 1.0
## log_lik[1459] 26 1.0
## log_lik[1460] 31 1.0
## log_lik[1461] 32 1.0
## log_lik[1462] 25 1.0
## log_lik[1463] 19 1.0
## log_lik[1464] 31 1.0
## log_lik[1465] 21 1.0
## log_lik[1466] 50 1.0
## log_lik[1467] 45 1.0
## log_lik[1468] 33 1.0
## log_lik[1469] 50 1.0
## log_lik[1470] 51 1.0
## log_lik[1471] 45 1.0
## log_lik[1472] 43 1.0
## log_lik[1473] 46 1.0
## log_lik[1474] 45 1.0
## log_lik[1475] 58 1.0
## log_lik[1476] 70 1.0
## psi[1] 107 1.0
## lp__ 11 1.2
##
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).ls(dfa.posterior)## [1] "data" "model" "monitor"
## [4] "samples" "samples_permuted"summary(dfa.posterior)## Length Class Mode
## model 1 stanfit S4
## samples_permuted 6 -none- list
## samples 148050 -none- numeric
## monitor 19740 -none- numeric
## data 1476 -none- numeric3.5 Diagrama de dependencia
Diagrama de dependencia del modelo de factores dinamico en medias
4 Especificacion dinamica en volatilidad
Seguimos considerando que los datos \((\boldsymbol{X}_t)_{t\geq 1}\) son \((p\times 1)\)-dimensionales y que existen \((\boldsymbol{f}_t)_{t\geq 1}\) vectores latentes \((k\times 1)\)-dimensionales, con \(k<<p\), tales que \[\begin{eqnarray} \boldsymbol{X}_t|\boldsymbol{f}_t & \sim & N_p(\boldsymbol{x}|\Lambda \boldsymbol{f}_t,\Sigma_t)\nonumber \\ \end{eqnarray}\] para \(t\geq 1\), donde \(\Lambda\) es una matriz de cargas de dimension \(p\times k\), y \[\Sigma_t=\text{diag}\{\exp(h_{1t}),\ldots,\exp(h_{pt})\},\] es la matriz de volatilidad estocastica, con \[\begin{eqnarray} \boldsymbol{f}_t|\boldsymbol{G}_{t} & \sim & N_k(\boldsymbol{f}|\boldsymbol{0},\boldsymbol{G}_t),\nonumber \end{eqnarray}\] considerando que \[\boldsymbol{G}_t=\text{diag}\{\exp(g_{1t}),\ldots,\exp(g_{kt})\}.\]
La estructura de dependencia entre las \(\Sigma_t\)s y las \(\boldsymbol{G}_t\)s se define separada y localmente como, \[\begin{eqnarray} h_{tj}|h_{(t-1)j} & \sim & N_1(h|\mu_{hj}+\phi_{hj}(h_{(t-1)j}-\mu_{hj}), s_{hj}),\nonumber \end{eqnarray}\] para \(j=1,\ldots,p\), y \[\begin{eqnarray} g_{ti}|g_{(t-1)i} & \sim & N_1(g|\mu_{gi}+\phi_{gi}(h_{(t-1)i}-\mu_{gi}), s_{gi}),\nonumber \end{eqnarray}\] para \(i=1,\ldots,k\), siendo las colecciones \((s_{hj})_{j=1}^p\) y \((s_{gi})_{i=1}^k\) escalares positivos conocidos.
4.1 Parametros
Asi, los parametros del modelo por estimar tornan en \[ \left(\Lambda,(\mu_{hj},\phi_{hj})_{j=1}^{p},(\mu_{gi},\phi_{gi})_{i=1}^{k}\right), \] y las variables latentes son \[ \left((\boldsymbol{f}_t)_{t\geq 1},(\boldsymbol{h}_t)_{t\geq 1},(\boldsymbol{g}_t)_{t\geq 1}\right) \approx \left((\boldsymbol{f}_t)_{t\geq 1},(\boldsymbol{\Sigma}_t)_{t\geq 1},(\boldsymbol{G}_t)_{t\geq 1}\right). \]
4.2 Observacion
Aunque \(\Lambda\) es fijo en \(t\), la estructura de codependencia de la \(p\) variables originales es reproducida por el modelo actual a traves de \(\boldsymbol{G}_t\), porque \[ \Omega_t=\Lambda\boldsymbol{G}_t\Lambda' + \boldsymbol{\Sigma}_t, \] para \(t\geq 1.\)
4.3 Implementacion
if(!require("factorstochvol")){install.packages("factorstochvol")}## Loading required package: factorstochvollibrary("factorstochvol")
datatsc.3 <- as.matrix(datatsc.3)
dsfa.posterior <- fsvsample(datatsc.3,
factors = 1,
draws = 150, burnin = 50, runningstore = 6)##
## Calling 1-factor MCMC sampler for 3 series of length 492.
##
##
********* Iteration 1 of 200 ( 0%) *********
********* Iteration 137 of 200 ( 68%) *********
********* Iteration 200 of 200 (100%) *********
##
## Reorganizing runningstores... Done!
## Ex-post sign-identification... Done!voltimeplot(dsfa.posterior)
corimageplot(dsfa.posterior, nrow(datatsc.3), plotCI = 'circle')
oldpar <- par(ask = FALSE)
plot(dsfa.posterior)
par(oldpar)4.4 Diagrama de dependencia
Diagrama de dependencia del modelo de factores dinamico en volatilidad
5 Comentarios
5.1 Generales
Los modelos flexibles que hemos visto en esta sesion presentan resultados
ad-hocque son confiables.Aunque confiables, estas soluciones no son generales y por lo menos dominantes a todas las circunstancias.
Hemos visto que cada implementacion tiene estructuras de
codificacion,usoynotacionparticulares. Estos aspectos siempre podran sortearse teniendo conocimiento de aspecto metodologico que involucra el modelo.En estas ilustraciones hemos corrido pocas simulaciones de las cadenas de Markov para hacer inferencia… En aplicaciones reales deberan correr mas, no demasiadas, de hecho.
5.2 Adecuacion del modelo
Hemos visto varias especificaciones del modelo de factores:
Estatico
Dinamico en nivel
Dinamico en nivel y volatilidad
Entre otras…
Adicionalmente, podemos considerar la capa de especificacion respecto a la reduccion de dimensionalidad.
Es asi pertinente ahora estudiar como cuantificar la adecuacion del modelo ante estas especificaciones.
5.2.1 Especificacion
Grosso modo, tendremos que para un conjunto de datos \[ \text{datos}=(\boldsymbol{x}_1,\ldots,\boldsymbol{x}_T), \] tedremos un conjunto de especificaciones \(m=1,\ldots,M\) (siendo \(M\) el numero todal de especificaciones), para las que podemos calcular la devianza del modelo, \(DIC(m)\), definido como \[ DIC(m)=-2 \widehat{\text{log-lik}}(\text{Modelo }m|\text{datos}) + 2\text{ep}(m), \] donde
\(\widehat{\text{log-lik}}(\text{Modelo }m|\text{datos})\) denota la logverosimilitud del modelo \(m\) evaluada en el estimador puntual (frecuentista o bayesiano) de los parametros del modelo \(m\), y
\(\text{ep}(m)\) denota el numero de parametros efectivos (medias y covarianzas) del modelo \(m\).
5.2.2 Calculo
A partir de las estimaciones via simulacion, podemos, en general, calcular los estimados puntuales de las medias, \[ \left\{\widehat{\boldsymbol{\mu}}_1,\ldots,\widehat{\boldsymbol{\mu}}_T\right\}, \] y de la estructura de covariabilidad, \[ \left\{\widehat{\boldsymbol{\Lambda}}_1,\ldots,\widehat{\boldsymbol{\Lambda}}_T\right\}, \] y \[ \left\{\widehat{\boldsymbol{\Sigma}}_1,\ldots,\widehat{\boldsymbol{\Sigma}}_T\right\}. \]
De manera general, la logverosimilitud del modelo \(m\) evaluada en los estimadores puntuales es calculada como \[ \widehat{\text{log-lik}}(\text{Modelo }m|\text{datos}) = \sum_{t=1}^{T}\text{log}\left\{ \text{N}_p\left(\boldsymbol{x}_t|\widehat{\boldsymbol{\mu}}_t,\widehat{\boldsymbol{\Omega}}_t\right)\right\}, \] donde \[ \widehat{\boldsymbol{\Omega}}_t =\widehat{\boldsymbol{\Lambda}}_t\widehat{\boldsymbol{\Lambda}}_1'+\widehat{\boldsymbol{\Sigma}}_t, \] para \(t=1,\ldots,T\).
Nota.- Si se trata de un modelo reducido (estructuralmente mas simple), entonces esta logverosimilitud puede simplificarse de acuerdo a tal estructura.
6 Lecturas complementarias
Jackson et al (2015) Specification and Estimation of Bayesian Dynamic Factor Models,
est46114_s13_suplemento1.pdfKastner et al (2017) Efficient bayesian Inference for Multivariate Factor Stochastic Volatility Models,
est46114_s13_suplemento2.pdfWest & Harrison (1997) Bayesian Forecasting and Dinamic Models, Springer, “Cap. Multivariate Modelling and Forecasting”.